THE  ELEMENTS 

OF 

THEORETICAL  AND  DESCRIPTIVE 

ASTEONOMY 

For  the  use  of  Colleges  and  Academies 


BY 

CHARLES  J.  WHITE,  A.M.: 

FORMERLY   PROFESSOR   OF   MATHEMATICS   OF   HARVARD   COLLEGE 

EIGHTH    EDITION,    REVISED 

By  PAUL  P.  BLACKBURN 

COMMANDER,  U.  8.  NAVY 


NEW  YORK 

JOHN  WILEY  &  SONS,  INC. 

LONDON:    CHAPMAN  &  HALL,    LIMITED 

1920 


Copyright,  1869,  1901, 

BY 
CHALES  J.  WHITE 


Copyright,  1920, 

BY 
PAUL  P    BLACKBURN 


PRESS  or 

BRAUNWORTH    &    CO. 

BOOK    MANUFACTURERS 

BROOKLYN.    N.    Y. 


PREFACE  TO  THE  EIGHTH  EDITION 


IN  revising  this  work  I  have  endeavored  to  bring  it  up 
to  date  without  making  material  changes  in  the  form, 
I  am  particularly  indebted  to  the  staff  of  the  Mount  Wilson 
Solar  Observatory  for  valuable  suggestions  and  for  assist- 
ance in  reading  the  proof,  and  to  the  Yerkes  Observatory 
for  photographs. 

P.  P.  B. 


429462 


PREFACE  TO  THE  FIFTH  EDITION 


THE  first  edition  of  this  work  was  published  in  1869,  to 
meet  the  requirements  of  the  students  of  the  United  States 
Naval  Academy,  In  preparing  it,  I  endeavored  to  present 
the  main  facts  and  principles  of  Astronomy  in  a  form 
adapted  to  the  elementary  course  of  instruction  in  that 
science  which  is  commonly  giyen  at  colleges  and  the  higher 
grades  of  academies.  I  selected  those  topics  which  seemed 
to  me  to  be  the  most  important  and  the  most  interesting, 
and  arranged  them  in  the  order  which  experience  had  led  me 
to  believe  to  be  the  best.  The  fifth  edition  of  the  book  is 
issued  with  no  change  in  the  general  plan,  and  with  only 
those  changes  which  the  advance  of  astronomical  knowledge 
in  the  last  few  years  renders  necessary. 

In  the  descriptive  portions  of  the  work,  I  have  endeavored 
to  give  the  latest  information  upon  every  topic  which  is 
introduced.  On  not  a  few  points  the  opinions  of  competent 
observers  are  by  no  means  the  same;  and  on  these  points  I 
have  endeavored  to  give,  as  far  as  possible,  the  various 
opinions  which  now  exist.  The  distances  and  the  dimen- 
sions of  the  heavenly  bodies  are  given  to  correspond  with 
the  value  of  the  solar  parallax  which  is  at  present  adopted 
in  the  American  Ephemeris;  the  recent  theories  upon  the 
connection  of  comets  and  meteors,  the  principles  of  spectro- 
scopic  observation,  and  the  conclusions  concerning  the  con- 
stitution and  the  movements  of  the  heavenly  bodies  which 
such  observation  induces,  are  given,  it  is  hoped,  in  sufficient 
detail.  The  NOTES  contain  some  of  the  latest  results. 


vi  PREFACE  TO  THE  FIFTH  EDITION 

No  clear  conception  of  the  processes  by  which  most  of 
the  fundamental  truths  of  Astronomy  have  been  established 
can  be  attained  without  some  knowledge  of  Mathematics. 
I  have  endeavored,  however,  to  confine  the  theoretic  dis- 
cussions within  the  limits  of  such  moderate  mathematical 
knowledge  as  may  fairly  be  expected  in  those  readers  for 
whom  the  treatise  is  intended.  Certain  definitions  and 
formulae,  with  which  the  student  may  possibly  not  be 
familiar,  will  be  found  in  the  Appendix;  and,  with  this  aid, 
I  believe  that  every  portion  of  the  work  can  be  read  without 
difficulty. 

In  the  preparation  of  the  work,  many  authorities  have 
been  consulted;  the  principal  ones  being  Chauvenet's 
Manual  of  Spherical  and  Practical  Astronomy,  and  Chambers' 
Descriptive  Astronomy.  In  obtaining  the  results  of  recent 
observations,  Professor  Newcomb's  two  treatises  have  been 
of  great  assistance  to  me.  The  treatise  has  been  used  as  a 
text-book  in  the  United  States  Naval  Academy,  the  Massa- 
chusetts Institute  of  Technology,  Harvard  College,  and  other 
institutions;  and  I  am  indebted  to  officers  of  these  institu- 
tions for  many  valuable  suggestions  as  to  errors  and  improve- 
ments. I  trust  that  this  new  edition  will  be  found  to  be 
free  from  mistakes,  and  that  it  will  be  useful,  not  only  to 
the  class  of  students  for  whom  it  is  especially  prepared, 
but  to  others  who  may  wish  to  know  the  general  principles 
and  the  present  state  of  the  science  of  Astronomy. 

Harvard  College, 
Cambridge,  Mass.,  1884. 


CONTENTS 


PAGE 

The  Greek  Alphabet xi 

CHAPTER  I 

GENERAL  PHENOMENA  OF  THE  HEAVENS.     DEFINITIONS.     THE 
CELESTIAL  SPHERE 

The  heavenly  bodies.  Astronomy.  Form  of  the  earth.  Diurnal 
motions  of  the  heavenly  bodies.  Right,  parallel,  and  oblique 
spheres.  Definitions.  Theorems.  The  astronomical  tri- 
angle. Spherical  co-ordinates.  Vanishing  lines  and  circles. 
Spherical  projections 1 

CHAPTER  II 

ASTRONOMICAL  INSTRUMENTS.     ERRORS 

The  clock:  its  error  and  rate.  The  chronograph.  The  astro- 
nomical telescope.  The  transit  instrument;  its  construction, 
adjustment,  and  use.  The  meridian  circle.  The  reading 
microscope.  Fixed  points.  The  altitude  and  azimuth  instru- 
ment. Method  of  equal  altitudes.  The  equatorial.  The 
spectroscope.  The  sextant.  The  artificial  horizon.  The 
vernier.  Other  astronomical  instruments.  Classes  of  errors.  21 

CHAPTER  III 

0 

REFRACTION.     PARALLAX.     DIP  OF  THE  HORIZON 

General  laws  of  refraction.     Astronomical  refraction.     Geocentric 

and  heliocentric  parallax.     The  dip  of  the  horizon 50 

CHAPTER  IV 
THE  EARTH.     ITS  SIZE,  FORM,  AND  ROTATION 

Measurement  of  arcs  of  the  meridian  by  triangulation.  Sphe- 
roidal form  of  the  earth.  Its  dimensions.  Its  volume,  density, 
and  weight.  Rotation  of  the  earth.  Change  of  weight  in 
different  latitudes.  Centrifugal  force.  The  trade-winds. 
Foucault's  pendulum  experiment.  Linear  velocity  of  rota- 
tion   60 

vii 


viii  CONTENTS 

CHAPTER  V 

LATITUDE  AND  LONGITUDE 

PAGE 

Four  methods  of  finding  the  latitude  of  a  place.  Latitude  at  sea. 
Reduction  of  the  latitude.  Longitude.  Greenwich  time  by 
chronometers  and  celestial  phenomena.  Difference  of  longi- 
tude by  electric  and  star  signals.  Longitude  at  sea.  Com- 
parison of  the  local  times  of  different  meridians 74 

CHAPTER  VI 

THE  SUN.     THE  EARTH'S  ORBIT.     THE  SEASONS.     TWILIGHT.     THE 
ZODIACAL  LIGHT 

The  ecliptic.  Distance  of  the  sun  from  the  earth  determined  by 
transits  of  Venus.  Magnitude  of  the  sun.  The  earth's  orbit 
about  the  sun.  The  seasons.  Twilight.  Rotation  of  the 
sun,  and  its  constitution.  The  zodiacal  light 86 

CHAPTER  VII 
SIDEREAL  AND  SOLAR  TIME.     EQUATION  OF  TIME.     THE  CALENDAR 

The  sidereal  and  the  solar  year.  Relation  of  sidereal  and  solar 
time.  The  equation  of  the  center.  The  equation  of  time. 
Astronomical  and  civil  time.  The  calendar 106 

CHAPTER  VIII 

UNIVERSAL  GRAVITATION.     PERTURBATIONS  IN  THE  EARTH'S  ORBIT. 

ABERRATION 

The  law  of  universal  gravitation.  The  mass  of  the  sun.  The 
earth's  motion  at  perihelion  and  aphelion.  Kepler's  laws. 
Precession.  Nutation.  Change  in  the  obliquity  of  the  ecliptic. 
Advance  of  the  line  of  apsides.  Diurnal  and  annual  aberra- 
tion. Velocity  of  light.  Aberration  a  proof  of  the  earth's 
revolution, ,,,,,,, 114 

CHAPTER  IX 

THE  MOON 

The  orbit  of  the  moon,  and  perturbations  in  it.  Variation  of  the 
moon's  meridian  zenith  distance.  Distance,  size,  and  mass 
of  the  moon.  Augmentation  of  the  semi-diameter.  The 
phases  of  the  moon.  Sidereal  and  synodical  periods.  Re- 


CONTENTS  ix 

PAGE 

tardation  of  the  moon.  The  harvest  moon.  Rotation, 
librations,  and  other  perturbations.  The  lunar  cycle.  Gen- 
eral description  of  the  moon 129 

CHAPTER  X 

LUNAR  AND  SOLAR  ECLIPSES.    OCCULTATIONS 

Lunar  eclipses.  The  earth's  shadow.  Lunar  ecliptic  limits. 
Solar  eclipses.  The  moon's  shadow.  Solar  ecliptic  limits. 
Cyclic  and  number  of  eclipses.  Occultations.  Longitude  by 
solar  eclipses  and  occultations 147 

CHAPTER  XI 

THE  TIDES 

Cause  of  the  tides.  Effect  of  the  moon's  change  in  declination. 
General  laws.  Influence  of  the  sun.  Priming  and  lagging  of 
tides.  The  establishment  of  a  port.  Cotidal  lines.  Height 
of  tides.  Tides  in  bays,  rivers,  etc.  Four  daily  tides.  Other 
phenomena 160 

CHAPTER  XII 
THE   PLANETS   AND   THE   PLANETOIDS.     THE   NEBULAR   HYPOTHESIS 

Apparent  motions  of  the  planets.  Heliocentric  parallax.  Orbits 
of  the  planets.  Inferior  planets.  Direct  and  retrograde 
motion.  Stationary  points.  Evening  and  morning  stars. 
Elements  of  a  planet's  orbit.  Heliocentric  longitude  of  the 
node.  Inclination  of  the  orbit.  Periodic  time.  Mercury. 
Venus.  Transits  of  Venus.  Superior  planets:  their  periodic 
times  and  distances.  Mars.  The  minor  planets.  Bode's 
law.  Jupiter:  its  belts,  satellites,  and  mass.  Saturn  and 
its  rings.  Disappearance  of  the  rings.  Uranus.  Neptune. 
The  nebular  hypothesis.  The  planetisimal  hypothesis 171 

CHAPTER  XIII 

COMETS  AND  METEORIC  BODIES 

General  description  of  comets.  The  tail.  Elements  of  a  comet's 
orbit.  Number  of  comets  and  their  orbits.  Periodic  times. 
Motion  in  their  orbits.  Mass  and  density.  Periodic  comets. 
Encke's  comet.  Winnecke's  or  Pons's  comet.  Brorsen's 
comet.  Biela's  comet.  D'Arrest's  comet.  Faye's  comet. 


CONTENTS 


Me"chain's  comet.  Halley's  comet.  Remarkable  comets 
of  the  19th  century.  The  great  comet  of  1811.  The  great 
comet  of  1843.  Donati's  comet.  The  great  comet  of  1861. 
The  great  comet  of  1882.  Meteoric  bodies.  Shooting  stars. 
The  November  showers.  Height  and  orbits  of  the  meteors. 
Detonating  meteors.  Aerolites.  Connection  of  comets  and 
meteoric  bodies. ...  .  208 


CHAPTER  XIV 

THE  FIXED  STARS.     NEBUL/E.     MOTION  OF  THE  SOLAR  SYSTEM.     REAL 
MOTIONS  OF  THE  STARS 

Proper  motions  of  the  fixed  stars.  Magnitudes.  Constellations. 
Constitution  of  the  stars.  Distance  of  the  stars.  Bessel's  dif- 
ferential observations.  Real  magnitudes  of  the  stars.  Vari- 
able and  temporary  stars.  Double  and  binary  stars.  Colored 
stars.  Clusters.  Resolvable  and  irresolvable  nebulae.  An- 
nular, elliptic,  spiral,  and  planetary  nebulae.  Nebulous  stars. 
Double  nebulae.  The  Magellanic  clouds.  Variation  of 
brightness  in  nebulae.  The  milky  way.  Number  of  the 
stars.  Motion  of  the  solar  system  in  space.  Real  motions 
of  stars  detected  with  the  spectroscope 240 

APPENDIX 

Mathematical  definitions,  theorems,  and  formulae 275 

Chronological  history  -of  astronomy 282 

Sketch  of  the  history  of  navigation 290 

Table        I. — Elements  of  the  planets,  the  sun,  and  the  moon ....  291 

II.— The  earth 293 

III.— The  moon 293 

' '         IV. — Elements  of  the  satellites 294 

V.— The  minor  planets 295 

' '         VI. — Schwabe's  observations  of  the  solar  spots 297 

' '        VII. — Transits  of  the  inferior  planets 298 

"      VIII.— The  constellations 299 

1 '         IX. — Examples  of  variable  stars 302 

' '          X. — Examples  of  binary  stars 303 

INDEX 305 


THE  GREEK  ALPHABET 


The  following  table  of  the  small  letters  of  this  alphabet 
is  given  for  the  use  of  those  readers  who  are  unacquainted 
with  the  Greek  language. 


a 

Alpha. 

V 

Nu. 

ft 

Beta. 

{ 

Xi. 

7 

Gamma. 

0 

Omicron. 

5 

Delta. 

7T 

Pi. 

6 

Epsilon. 

P 

Rho. 

r 

Zeta. 

<7 

Sigma. 

y 

Eta. 

r 

Tau. 

#  or  0 

Theta 

V 

Upslloa 

i 

Iota. 

0 

Phi. 

K 

Kappa. 

X 

Chi 

X 

Lambda. 

# 

Psi. 

M 

Mu. 

CO 

Omega. 

XI 


ASTRONOMY 


CHAPTER  I 
GENERAL  PHENOMENA  OF  THE  HEAVENS.    DEFINITIONS 

1.  The  heavenly  bodies  are  the  sun,  the  planets,  the 
satellites    of    the    planets,    the    comets,    the   meteors,   the 
nebula?,  and  the  fixed  stars. 

The  planets  revolve  about  the  sun  in  elliptical  orbits,  and 
the  satellites  revolve  in  similar  orbits  about  the  planets. 
The  earth  is  a  planet,  as  we  shall  see  hereafter,  and  the 
moon  is  its  satellite.  The  comets  revolve  about  the  sun  in 
orbits  which  are  either  ellipses,  parabolas,  or  hyperbolas. 
Comparatively  little  is  known  with  any  degree  of  certainty 
about  the  meteors:  but  it  is  probable  that  they  too  revolve 
about  the  sun. 

The  sun,  the  planets,  the  satellites,  and  the  comets  con- 
stitute what  is  called  the  solar  system.  The  fixed  stars  are 
bodies  which  lie  outside  of  this  system,  and  preserve  almost 
precisely  the  same  configuration  from  year  to  year. 

The  heavenly  bodies  may  be  considered  to  be  projected 
upon  the  concave  surface  of  a  sphere  of  indefinite  radius, 
the  eye  of  the  observer  being  at  the  center  of  the  sphere. 
This  sphere  is  called  the  celestial  sphere. 

2.  Astronomy  is  'the  science  which  treats  of  the  heavenly 
bodies.     It  may  be  divided  into  Theoretical,  Practical,  and 
Descriptive  Astronomy. 


2  ^ L  :  GENERAL  PHENOMENA  OF  THE  HEAVENS 

Theoretical  Astronomy  may  be  divided  into  Spherical 
and  Physical  Astronomy. 

Spherical  Astronomy  treats  of  the  heavenly  bodies  when 
considered  to  be  projected  upon  the  surface  of  the  celestial 
sphere.  It  embraces  those  problems  which  arise  from  the 
apparent  diurnal  motion  of  the  heavenly  bodies,  and  also 
those  which  arise  from  any  changes  in  the  apparent  positions 
of  these  bodies  upon  the  surface  of  the  celestial  sphere. 

Physical  Astronomy  treats  of  the  causes  of  the  motions 
of  the  heavenly  bodies,  and  of  the  laws  by  which  these 
motions  are  governed. 

Practical  Astronomy  treats  of  the  construction,  adjust- 
ment and  use  of  astronomical  instruments. 

Descriptive  Astronomy  includes  a  general  description  of 
the  heavenly  bodies;  of  their  magnitudes,  distances,  motions, 
and  configuration;  of  their  appearance  and  structure;  of,  in 
short,  everything  relating  to  these  bodies  which  comes  from 
observation  or  calculation. 

3.  Form  of  the  Earth. — We  may  assume,  at  the  outset, 
that  the  form  of  the  earth  is  very  nearly  that  of  a  sphere. 
The  following  are  some  of  the  reasons  which  may  be  given 
for  such  an  assumption: 

(1)  If  we  stand  upon  the  sea-shore,  and  watch  a  ship 
which  is  receding  from  the  land,  we  shall  find  that  the  top- 
masts remain  in  sight  after  the  hull  has  disappeared.  If  the 

surface  of  the  sea  were 
merely  an  extended 
plane,  this  would  not 
happen;  for  the  top- 
masts,  being  smaller  in 
dimensions  than  the 
hull,  would  in  that  case 

FlG-  L  disappear    first.      The 

supposition     that     the 

surface  of  the  sea  is   curved,  however,  fully  accounts  for 
this  phenomenon,  as    may  be    seen    in  Fig.    1.     Let  the 


DIURNAL  MOTION  OF  THE  HEAVENLY  BODIES        3 

curve  CBG  represent  a  portion  of  the  earth's  surface, 
and  let  A  be  the  position  of  the  observer's  eye:  it  is  at 
once  evident  that  no  portion  of  the  ship,  St  will  be  visible 
which  is  situated  below  the  line  A  H,  drawn  from  A  tangent 
to  the  earth's  surface  at  C.  The  same  figure  also  shows 
why  it  is  that  when  a  ship  is  approaching  land  any  object 
on  shore  can  be  seen  from  the  topmasts  before  it  is  seen 
from  the  deck. 

(2)  At  sea,  the  visible  horizon  everywhere  appears  to  be 
a  circle.     This  also  is  easily  explained  on  the  supposition 
that  the  earth  is  spherical  in  form;  for  if,  in  Fig.  1,  the  line 
A  H  is  turned  about  the  point  A,  and  is  continually  tangent 
to  the  sphere,  the  points  of  tangency,  C,  D,  E,  etc.,  will  form 
the  visible  horizon  of  the  spectator,  and  will  evidently  con- 
stitute a  circle. 

(3)  A  lunar  eclipse  occurs  when  the  earth  is  situated 
between  the  moon  and  the  sun.     Now  the  shadow  which 
the  earth  at  such  a  time  casts  upon  the  moon  is  invariably 
circular  in  form :  and  a  body  which  in  every  position  casts  a 
circular  shadow  must  be  a  sphere. 

4.  Diurnal  Motion  of  the  Heavenly  Bodies. — Two  things 
v/ill  be  noticed  by  an  observer  who  watches  the  heavens 
during  any  clear  night.  The  first  is,  that  all  the  heavenly 
bodies,  with  the  exception  of  the  moon  and  the  planets, 
retain  constantly  the  same  relative  situation;  and  the  second 
is,  that  all  these  bodies,  without  any  exception  whatever, 
are  continually  changing  their  positions  with  reference  to  the 
horizon.  Let  us  suppose  the  observer  to  be  at  some  place 
in  the  Northern  Hemisphere.  A  plane  passed  tangent  to  the 
earth's  surface  at  his  feet  will  be  the  plane  of  his  sensible 
horizon,  and  a  second  plane,  parallel  to  this,  passed  through 
the  center  of  the  earth,  will  be  that  of  his  rational  horizon. 
If  these  two  planes  be  indefinitely  extended  in  every  direction, 
they  will  intersect  the  surface  of  the  celestial  sphere  in  two 
circles;  but  the  radius  of  the  celestial  sphere  is  so  immense 
in  comparison  with  the  radius  of  the  earth,  that  these  two 


4  GENERAL  PHENOMENA  OF  THE  HEAVENS 

circles  will  sensibly  coincide,  and  will  form  one  great  circle 
of  the  sphere,  called  the  celestial  horizon.  In  other  words, 
the  earth,  when  compared  with  the  celestial  sphere,  is  to  be 
regarded  as  only  a  point  at  its  center. 

Let  Fig.  2,  then,  represent  the  celestial  sphere,  at  the 
center  of  which,  0,  the  observer  is  stationed.  Let  the  circle 
HESW  be  his  celestial  horizon,  of  which  H  is  the  north 
point,  S  the  south,  E  the  east,  and  W  the  west. 


If  he  looks  towards  the  southern  point  of  the  horizon,  and 
watches  the  movements  of  some  star  which  rises  a  little  east 
of  south,  he  will  see  that  it  rises  above  the  horizon  in  a  cir- 
cular path  for  a  little  distance,  attains  its  greatest  elevation 
above  the  horizon  when  it  bears  directly  south,  and  then 
descends  and  passes  below  the  horizon  a  little  west  of  south. 
In  the  figure,  abc  represents  the  path  of  such  a  star.  If  he 
notices  a  star  which  rises  more  to  the  eastward,  as  at  d,  for 
instance,  he  will  see  that  it  also  passes  from  the  east  quarter 
of  the  horizon  to  the  west  in  a  circular  path,  attaining, 


INFERENCES  3 

however,  a  greater  altitude  above  the  horizon  than  that 
which  the  star  a  attained,  and  remaining  above  the  horizon 
a  longer  time.  A  star  which  rises  in  the  east  point  will  set 
in  the  west  point,  and  will  remain  twelve  hours  above  the 
horizon.  Turning  his  attention  to  some  star  which  rises 
between  the  north  and  the  east,  as  the  star  g,  for  instance, 
he  will  find  that  its  movements  are  similar  to  the  movements 
of  the  stars  already  noticed,  and  that  it  will  remain  above 
the  horizon  for  more  than  twelve  hours.  Finally,  if  he  turns 
towards  the  north,  he  will  see  certain  stars,  called  circum- 
polar  stars,  which  never  pass  below  the  horizon,  but  con- 
tinually revolve  about  a  fixed  point  in  the  heavens,  very 
near  to  which  point  is  a  bright  star  called  the  Pole-star, 
which,  to  the  naked  eye,  appears  to  be  stationary,  though 
observation  shows  that  it  also  revolves  about  this  same  fixed 
point.  In  the  figure  Im  represents  the  orbit  of  a  circum- 
polar  star. 

If  the  same  course  of  observations  be  repeated  on  the 
following  night,  the  observer  will  find  that  the  situations  of 
the  stars  with  reference  both  to  each  other  and  to  the  horizon 
are  the  same  that  they  were  when  he  first  began  to  examine 
them;  the  motions  which  have  already  been  described  will 
be  repeated,  the  circumpolar  stars  will  still'  revolve  about 
the  same  point  in  the  heavens,  and,  in  short,  all  the  phe- 
nomena of  which  he  took  note  will  again  be  exhibited. 

5.  Inferences. — Three  important  truths  are  proved  by 
a  series  of  observations  similar  to  that  which  we  suppose  to 
have  been  made. 

(1)  The  points  at  which  the  path  of  each  star  intersects 
the  horizon  remain  unchanged  from  night  to  night,  as  long  as 
the  geographical  position  of  the  observer  remains  the  same. 

(2)  All  the  stars,  whether  they  move  in  great  or  in  small 
circles,  make  a  complete  revolution  in  identically  the  same 
interval  of  time; — that  is  to  say,  in  twenty-four  sidereal 
hours. 

(3)  If  we  call  that  point  about  which  the  circumpolar 


6     GENERAL  PHENOMENA  OF  THE  HEAVENS 

stars  appear  to  revolve  the  north  pole  of  the  heavens,  and 
call  the  right  line  drawn  from  this  point  through  the  common 
center  of  the  earth  and  the  celestial  sphere  the  axis  of  the 
celestial  sphere,  the  planes  of  the  circles  of  all  the  stars  are 
perpendicular  to  this  axis. 

Whether,  then,  the  earth  remains  at  rest,  and  the  celestial 
sphere  rotates  about  its  axis,  as  above  defined,  or  the  celestial 
sphere  remains  at  rest,  and  the  earth  rotates  within  it  on  an 
axis  of  its  own,  one  thing  is  certain:  The  axis  of  rotation 
preserves  in  either  case  a  constant  direction.  If  the  celestial 
sphere  rotates  about  its  axis,  this  axis  always  passes  through 
the  same  points  of  the  earth's  surface;  and  if  the  earth 
rotates  within  the  celestial  sphere,  its  axis  of  rotation  is 
constantly  directed  to  the  same  points  on  the  surface  of  the 
celestial  sphere. 

We  shall  see  hereafter  the  reasons  which  have  led  to  the 
adoption  of  the  theory  that  these  apparent  motions  of  the 
stars  are  really  due  to  the  rotation  of  the  earth  upon  its  own 
axis.  At  present,  for  the  sake  of  convenience  in  description, 
we  shall  consider  the  earth  to  be  at  rest,  and  shall  speak  of 
the  apparent  motions  of  the  heavenly  bodies  as  though  they 
were  real. 

6.  Further  Observations. — Let  us  now  suppose  that  the 
observer  leaves  the  place  where  he  has  hitherto  been  sta- 
tioned, and  travels  in  the  direction  of  the  point  about  which 
the  circumpolar  stars  have  appeared  to  revolve,  and  which 
we  have  called  the  north  pole  of  the  heavens.  The  general 
character  of  the  phenomena  which  he  observes  will  not  be 
changed;  but  he  will  notice  a  change  in  this  respect:  the 
elevation  of  the  north  pole  above  the  horizon  will  continually 
increase  as  he  travels  towards  it,  and  the  planes  of  the 
circles  of  the  stars,  remaining  constantly  perpendicular  to 
the  axis  of  the  celestial  sphere,  will  become  less  and  less 
inclined  to  the  plane  of  the  horizon.  The  consequence  of 
this  will  be  that  stars  which  are  near  the  southern  point  of 
the  horizon  will  remain  a  shorter  time  above  the  horizon, 


FURTHER  OBSERVATIONS  7 

and  will  finally  cease  to  appear;  while  in  the  northern 
quarter  of  the  heavens  the  number  of  stars  which  never  pass 
below  the  horizon  will  continually  increase.  Finally,  if  we 
suppose  the  observer  to  go  on  until  the  north  pole  is  directly 
above  his  head,  the  stars  which  he  sees  will  neither  set  nor 
rise,  but  will  continually  move  about  the  sphere  in  circles 
whose  planes  are  parallel  to  the  plane  of  the  horizon.  In 
such  a  situation,  it  is  evident  that  half  of  the  celestial  sphere 
will  be  perpetually  invisible  to  him.  Referring  to  Fig.  2, 
such  a  state  of  things  is  represented  by  supposing  the  line  OP 
to  be  moved  up  into  coincidence  with  OZ,  the  planes  of  the 
circles  abc,  def,  etc.,  still  remaining  perpendicular  to  OP. 
The  stars  a  and  d  will  lie  continually  below  the  horizon,  and 
the  stars  g  and  /  continually  above  it. 

Such  a  sphere  as  this  just  now  described,  where  the  planes 
of  revolution  are  parallel  to  the  plane  of  the  horizon,  is 
called  a  parallel  sphere. 

If  the  observer,  instead  of  traveling  towards  the  north 
pole,  travels  directly  from  it,  its  elevation  above  the  horizon 
will  continually  decrease,  and  the  obliquity  of  the  planes 
of  the  circles  to  the  plane  of  the  horizon  will  continually 
increase,  until  he  will  at  length  reach  a  point  at  which  the 
north  pole  will  lie  in  his  horizon  and  the  planes  of  the  circles 
will  be  perpendicular  to  its  plane.  Referring  again  to 
Fig.  2,  the  line  PO  will  in  this  case  coincide  with  OH,  and 
the  arcs  abc,  def,  etc.,  will  have  their  planes  perpendicular 
to  the  plane  of  the  horizon.  It  is  evident  that  at  this  point 
every  star  in  the  celestial  sphere  will  come  above  the  horizon 
once  in  twenty-four  hours,  and  that  half  of  every  circle  will 
lie  above  the  horizon,  and  half  below  it. 

The  geographical  position  which  the  observer  has  now 
reached  is  some  point  on  the  earth's  equator  (Art.  7).  Such 
a  sphere  as  this,  where  the  planes  of  the  circles  are  per- 
pendicular to  the  plane  of  the  horizon,  is  called  a  right 
sphere.  Besides  the  right  and  the  parallel  sphere,  we  have 
also  the  oblique  sphere,  where  the  planes  of  the  circles  are 


8      GENERAL  PHENOMENA  OF  THE  HEAVENS 

oblique  to  the  plane  of  the  horizon.     Such  a  sphere  is  repre- 
sented in  Fig.  2. 

If  the  observer  travels  still  farther  in  the  same  direction, 
the  north  pole  will  sink  below  his  horizon  and  the  other 
extremity  of  the  axis  of  the  celestial  sphere,  called  the  south 
pole,  will  rise  above  it.  There  will  be  circumpolar  stars 
revolving  about  this  pole,  and,  in  brief,  all  the  phenomena 
which  the  observer  noticed  while  traveling  towards  the 
north  pole  will  be  repeated  as  he  travels  towards  the  south 
pole.* 

DEFINITIONS 

7.  We  are  now  prepared  to  define  certain  points,  angles, 
and  circles  on  the  earth  and  on  the  celestial  sphere. 

The  axis  of  the  earth  is  that  diameter  of  the  earth  about 
which  it  revolves.  The  intersections  of  this  axis  with  the 
surface  of  the  earth  are  the  north  pole  and  the  south  pole. 

The  indefinite  prolongation  of  the  earth's  axis  is  the 
axis  of  the  celestial  sphere  and  it  is  about  this  axis,  as  we 
have  already  seen,  that  the  celestial  bodies  appear  to  revolve. 
Where  this  imaginary  line  meets  the  celestial  sphere  are  the 
celestial  poles.  That  pole  of  the  heavens  which  is  above  the 
horizon  at  any  place  is  called  the  elevated  pole  at  that  place; 
the  other  is  called  the  depressed  pole. 

The  earth's  equator  is  that  great  circle  of  the  earth 
whose  plane  is  perpendicular  to  the  axis.  This  circle  is,  of 
course,  equidistant  from  the  two  poles,  and  its  plane  divides 
the  earth  into  two  hemispheres.  That  hemisphere  which 
contains  the  north  pole  is  called  the  northern  hemisphere, 
the  other  is  called  the  southern  hemisphere. 

Parallels  of  latitude  are  small  circles  of  the  earth  whose 
planes  are  perpendicular  to  the  axis  and,  consequently 
parallel  to  the  plane  of  the  equator. 

Terrestrial  meridians  are  great  circles  of  the  earth  passing 
through  the  poles;  it  is  evident  that  the  planes  of  the 


DEFINITIONS  9 

meridians  are  perpendicular  to  the  plane  of  the  equator  and 
that  the  earth's  axis  lies  in  all  of  these  planes. 

The  latitude  of  any  place  on  the  earth's  surface  is  its 
angular  distance  from  the  plane  of  the  earth's  equator, 
measured  by  the  arc  of  the  meridian  included  between  the 
place  and  the  equator.  Latitude  is  reckoned  either  north 
or  south  of  the  equator,  from  0°  to  90°. 

The  longitude  of  any  place  on  the  earth's  surface  is  the 
inclination  of  its  own  meridian  to  the  meridian  of  some 
fixed  station,  and  is  measured  by  the  arc  of  the  equator 
included  between  these  two  meridians.  Longitude  is 
reckoned  east  or  west  of  the  fixed  meridian,  from  0°  to  180°. 
The  meridian  of  Greenwich,  England,  is  taken  for  the  fixed 
meridian  by  most  nations,  including  the  United  States. 
The  fixed  meridian  is  called  the  prime  meridian. 

The  linear  distance  between  any  two  meridians  measured 
on  the  arc  of  a  parallel  between  those  meridians  is  the 
departure  between  those  meridians  for  that  latitude.  It  is 
evident  that  if  the  departure  between  any  two  meridians  is 
taken  on  two  different  parallels,  the  departure  at  the  higher 
latitude  will  be  the  smaller. 

Upward  motion  at  any  place  is  motion  from  the  center 
of  the  earth;  downward  motion  is  toward  the  same  point. 

The  celestial  horizon  has  already  been  defined  (Art.  4). 
The  vertical  line  is  a  straight  line  perpendicular  to  the  plane 
of  the  horizon  at  the  observer.  The  two  points  in  which 
this  line,  indefinitely  prolonged,  meets  the  celestial  sphere 
are  the  zenith  and  nadir,  the  first  above  and  the  second 
below  the  horizon. 

The  celestial  meridian  of  any  place  is  the  great  circle  in 
which  the  plane  of  the  terrestrial  meridian  of  that  place, 
when  indefinitely  produced,  meets  the  surface  of  the  celestial 
sphere.  The  celestial  axis  divides  the  celestial  meridian 
into  two  semi-circumferences;  that  which  is  on  the  same 
side  of  the  axis  as  the  zenith  is  the  upper  branch  of  the 
meridian  and  the  other  is  called  the  lower  branch.  The 


10     GENERAL  PHENOMENA  OF  THE  HEAVENS 

points  where  the  celestial  meridian  and  the  celestial  horizon 
intersect  are  called  the  north  and  the  south  point  of  the  hori- 
zon,— the  point  which  is  the  nearer  to  the  north  pole  being 
the  north  point.  The  line  in  which  the  planes  of  these  same 
two  great  circles  intersect  is  called  the  meridian  line. 

Vertical  circles  are  great  circles  of  the  sphere  which  pass 
through  the  zenith  and  nadir.  That  vertical  circle  the  plane 
of  which  is  perpendicular  to  the  plane  of  the  celestial  meridian 
is  called  the  prime  vertical.  The  points  in  which  the  prime 
vertical  cuts  the  horizon  are  called  the  east  and  the  west  point 
of  the  horizon;  and  the  line  which  joins  these  two  points 
is  the  east  and  west  line. 

The  celestial  equator,  also  called  the  equinoctial,  is  the  great 
circle  of  the  sphere  in  which  the  plane  of  the  earth's  equator, 
indefinitely  produced,  meets  the  celestial  sphere. 

The  altitude  of  a  heavenly  body  is  its  angular  distance 
above  the  plane  of  the  celestial  horizon,  measured  on  a 
vertical  circle  passing  through  that  body.  The  zenith 
distance  of  the  body  is  its  angular  distance  from  the  zenith, 
and  is  evidently  the  complement  of  the  altitude. 

The  azimuth  of  a  celestial  body  is  the  inclination  of  the 
vertical  circle  which  passes  through  the  body  to  the  celestial 
meridian,  and  is  measured  by  the  arc  of  the  celestial  horizon 
included  between  this  vertical  circle  and  the  celestial 
meridian.  Azimuth  may  be  reckoned  from  either  the  north 
or  the  south  point  of  the  horizon,  and  towards  either  the 
west  or  the  east.  Navigators  usually  reckon  it  from  the 
north  point  to  the  right  hand,  or  clock-wise,  from  0°  to  360°, 
though  it  may  be  reckoned  from  the  north  point  in  north 
latitude  and  the  south  point  in  south  latitude,  and  east  or 
west  of  the  meridian,  thus  restricting  it  numerically  to  values 
less  than  180°.  This  latter  method  is  the  one  usually 
employed  in  tables  of  azimuths.  Astronomers,  on  the 
contrary,  usually  reckon  from  the  south  point  to  the  right, 
from  0°  to  360°. 

The  amplitude  of  a  heavenly  body  is  its  angular  distance 


DEFINITIONS  11 

from  the  prime  vertical  when  in  the  horizon.  It  is  reckoned 
from  the  east  point  when  the  body  is  rising,  and  from  the 
west  point  when  the  body  is  setting,  towards  the  north  or 
the  south  as  the  body  is  to  the  north  or  the  south  of  the 
prime  vertical. 

Hour  circles,  or  circles  of  declination,  are  great  circles  of 
the  celestial  sphere  passing  through  the  poles  of  the 
heavens. 

The  declination  of  a  heavenly  body  is  its  angular  distance 
from  the  plane  of  the  celestial  equator,  measured  on  the 
hour  circle  passing  through  the  body.  It  is  reckoned  in 
degrees,  minutes,  and  seconds,  to  the  north  and  the  south. 
The  polar  distance  of  a  body  is  its  angular  distance  from 
either  pole,  measured  on  its  hour  circle.  Usually,  however, 
when  we  speak  of  the  polar  distance  of  a  body,  we  mean 
its  angular  distance  from  the  elevated  pole. 

If,  in  Fig.  2,  with  the  arc  HP,  which  measures  the 
altitude  of  the  elevated  pole,  as  a  polar  radius,  we  describe 
a  circle  about  the  pole  as  a  center,  it  is  evident  that  the 
stars  whose  circles  lie  within  this  circle  will  never  set.  This 
circle  is  called  the  circle  of  perpetual  apparition.  It  is  equally 
evident  that  stars  whose  circles  lie  within  a  circle  of  the 
same  magnitude  described  about  the  depressed  pole  as  a 
center,  will  never  come  above  the  horizon.  This  circle  is 
therefore  called  the  circle  of  perpetual  occultation. 

The  passage  of  a  celestial  body  across  the  meridian  is 
called  its  transit  or  culmination.  When  the  body  is  within 
the  circle  of  perpetual  apparition,  both  transits  occur  above 
the  horizon,  one  above  the  pole  and  the  other  below  it. 
These  are  called  the  upper  and  the  lower  transit.  For  all 
bodies  outside  this  circle,  and  not  within  the  circle  of  per- 
petual occultation,  the  upper  transit  occurs  above  the 
horizon,  the  lower  below  it.  For  all  bodies  whatever,  the 
upper  transit  occurs  when  the  body  crosses  the  upper  branch 
of  the  meridian,  and  the  lower  transit  when  it  crosses  the 
lower  branch. 


12     GENERAL  PHENOMENA  OF  THE  HEAVENS 

The  vernal  equinox  is  a  certain  .fixed  point  upon  the 
equinoctial.  It  is  also  called  the  first  paint  of  Aries. 

The  right  ascension  of  a  heavenly  body  is  the  inclination 
©f  its  hour  circle  to  the  hour  circle  which  passes  through  the 
vernal  equinox;  or  it  is  the  arc  of  the  equinoctial  inter- 
cepted between  these  two  hour  circles.  Right  ascension  is 
reckoned  in  hours,  minutes,  and  seconds  (an  hour  being 
taken  equal  to  15°  of  arc),  to  the  eastward  from  the  vernal 
equinox,  from  Oh.  to  24h. 

The  hour  angle  of  a  heavenly  body  is  the  inclination  of 
the  hour  circle  which  passes  through  the  body  to  the 
celestial  meridian,  and  is  measured  by  the  arc  of  the  celestial 
equator  included  between  these  two  circles.  Hour  angles  are 
reckoned  positively  towards  the  west,  from  the  upper  culmi- 
nation, from  0°  to  360°,  or  Oh.  to  24h. 

The  hour  angle  of  the  sun  is  called  solar  time,  and  that  of 
the  first  point  of  Aries  sidereal  time.  The  interval  of  time 
between  two  consecutive  upper  transits  of  the  sun  is  called 
a  solar  day,  and  the  interval  between  the  upper  transits  of 
the  first  point  of  Aries  is  called  a  sidereal  day.  The  celestial 
sphere  apparently  makes  one  revolution  about  the  earth  in 
a  sidereal  day.  The  solar  day  is,  on  the  average,  about  3m. 
56s.  of  solar  time  longer  than  the  sidereal  day.  a 

8.  Some  of  the  preceding  definitions  are  illustrated  in 
the  diagram,  Fig.  3.  In  this  figure  0  is  the  position  of  the 
observer,  HESW  his  celestial  horizon,  Pp  the  axis  of 
the  heavens,  P  the  elevated  and  p  the  depressed  pole,  Z  the 
zenith,  and  N  the  nadir.  The  circle  HZSN  is  the  observer's 
celestial  meridian.  It  may  be  noticed  that  this  circle  is  at 
once  a  vertical  and  an  hour  circle.  The  circle  ECWD  is 
the  equinoctial,  and  the  circle  EZWN,  perpendicular  to  the 
meridian,  is  the  prime  vertical,  cutting  the  horizon  in  E 
and  W,  the  east  and  the  west  point  of  the  horizon.  The 
equinoctial,  being  also  perpendicular  to  the  meridian,  passes 
through  the  same  points.  If  P  is  supposed  to  be  the  north 
pole,  H  is  the  north  point  of  the  horizon,  and  S  the  south  point. 


\       I 


DEFINITIONS 


13 


Let  A  denote  some  celestial  body.  GA  is  its  altitude, 
ZA  its  zenith  distance,  HESG  its  azimuth  as  reckoned  by 
navigators,  and  SG  its  azimuth  as  reckoned  by  astronomers, 
all  these  elements  of  position  being  determined  by  the  arc 
of  a  vertical  circle,  ZG,  passed  through  A.  Let  an  arc  of 
an  hour  circle,  PB,  be  also  passed  through  A.  Then  is 
AB  its  declination,  PA  its  polar  distance,  and  if  V  be 
taken  to  denote  the  position  of  the  vernal  equinox,  VB 


is  the  right  ascension  of  A.  The  right  ascension  may  also 
be  represented  by  the  angle  VPB,  which  the  arc  VB 
measures.  The  angle  ZPB  is  the  hour  angle  of  A,  and 
ZPV  is  the  hour  angle  of  the  vernal  equinox,  or  the  sidereal 
time.  This  angle  may  also  be  designated  as  the  right 
ascension  of  the  meridian. 

The  circle  K  H,  drawn  about  P  with  the  radius  P  H,  is 
the  circle  of  perpetual  apparition.  The  star  whose  path  is 
represented  by  Im  never  passes  below  the  horizon:  I  is  its 


14 


GENERAL  PHENOMENA  OF  THE  HEAVENS 


upper,  m  its  lower  culmination.  SR  represents  the  circle  of 
perpetual  occupation,  and  the  stars  whose  paths  lie,  like  or 
within  this  circle,  never  come  above  the  horizon. 

9.  Theorem. — The  sidereal  time  at  any  place  is  always 
equal  to  the  sum  of  the  right  ascension  and  the  hour  angle  of 
the  same  body.     This  is  an  important  astronomical  theorem, 
and  is  readily  proved  in  Fig.  3,  in  which  ZPV,  the  sidereal 
time,  is  the  sum  of  ZPB,  the  hour  angle,  and  VPB,  the 
right  ascension  of  the  celestial  body  A.     Any  two  of  these 
three  angles,  then,  being  given,  the 'third  is  readily  obtained. 

Corollary. — When  a  celestial  body  is  at  its  upper  culmina- 
tion at  any  place,  its  right  ascension  is  equal  to  the  sidereal 
time  at  that  place.  This  is  evidently  true,  because  the  hour 
angle  of  a  body  when  at  its  upper  culmination  is  zero,  and 
hence,  from  the  theorem,  the  right  ascension  of  the  body  is 
equal  to  the  sidereal  time.  For  example,  in  Fig.  3,  if  /  is  a 
body  at  its  upper  culmination,  VPC  is  both  its  right  ascen- 
sion and  the  right  ascension  of  the  meridian. 

10.  Theorem. — The  latitude  of  any  place  on  the  earth's 
surface  is  equal  to  the  altitude  of  the  elevated  pole  at  that  place. 

Let  L  (Fig.  4)  be  some 
place  on  the  earth's  sur- 
face, Pp  the  earth's  axis, 
and  EQ  the  equator. 
The  line  HR,  tangent 
to  the  earth's  surface  at 
L,  is  the  horizon,  and  Z 
the  zenith  of  L.  •  Accord- 
Ing  to  the  definition  al- 
ready given,  LOQ  is  the 
latitude  of  L.  Let  the 
earth's  axis  and  the  plane 
of  the  earth's  equator 

be  indefinitely  prolonged,  and  •  at  L  let  the  line  LP", 
parallel  to  the  earth's  axis,  be  also  indefinitely  prolonged. 
Owing  to  the  immensity  of  the  celestial  sphere  when  com- 


FIG.  4. 


THE  ASTRONOMICAL  TRIANGLE  15 

pared  with  the  earth,  these  two  lines  will  sensibly  meet  at  a 
common  point  on  the  surface  of  the  celestial  sphere,  and  this 
common  point  will  be  the  elevated  pole.  The  elevated 
pole,  then,  to  an  observer  at  L,  will  lie  in  the  direction  LP", 
and  P"LH  will  be  its  altitude. 

Now  we  have,  HLZ  =  POQ 

ZLP"=ZOP' 

.:  P"LH  =  LOQ 
which  was  to  be  proved. 

11.  Theorem. —  The  latitude  of  a  place  is  equal  to  the 
declination  of  the  zenith  of  that  place. 

The  latitude  of  the  place  L  is  measured  by  the  angle 
LOQ  and  the  declination  of  the  zenith  of  that  place  is 
measured  by  the  angle  ZOQ'.  Since  these  angles  are  the 
same  the  theorem  is  proved. 

Either  of  these  theorems  might  be  deduced  from  the 
other,  but  it  is  better  to  consider  them  as  independent 
propositions. 

12.  The  Astronomical  Triangle. — The  spherical  triangle 
PZA  (Fig.  3)  is  called  the  astronomical  triangle.     It  is  formed 
by  the  arcs  of  the  meridian  of  the  place,  and  of  the  vertical 
circle  and  the  hour  circle  passing  through  some  heavenly 
body,  which  are  included  between  the  zenith  of  the  observer, 
the  elevated  pole,  and  the  position  of  the  body  as  projected 
on  the  surface  of  the  celestial  sphere.     The  three  sides  are: 
ZP,  the  co-latitude  of  the  place;   PA,  the  polar  distance  of 
the  body;   and  ZA,  its  zenith  distance.     The  three  angles 
are:  ZPA,  the  hour  angle  of  the  body;   PZA,  its  azimuth; 
and  PAZ,  an  angle    which    is   rarely  used,  and  which  is 
commonly  called  the  position  angle  of  the  body. 

The  co-latitude  and  the  zenith  distance  can  evidently 
never  be  greater  than  90°.  The  polar  distance  is  equal  to 
90°  minus  the  declination:  and  it  is  less  than  90°  when  the 
body  is  on  the  same  side  of  the  celestial  equator  as  the 


16     GENERAL  PHENOMENA  OF  THE  HEAVENS 

observer,  but  becomes  numerically  90°  plus  the  declination 
when  the  body  is  on  the  opposite  side.  In  the  former  case 
the  declination  of  the  body  is  said  to  have  the  same  name  as 
the  latitude,  in  the  latter  case,  to  have  the  opposite  name. 

13.  Diurnal  Circles. — We  have  already  seen  that  the 
apparent  daily  motions  of  the  stars  are  performed  in  circles, 
the  planes  of  which  are  perpendicular  to  the  axis  of  the  sphere. 
These  circles  are  called  diurnal  circles.     The  phenomena 
which  have  been  observed  with  reference  to  these  circles 
(Arts.  4  and  6)  are  explained  in  Fig.  3.     The  circles  of  all 
the  stars  which  rise  in  the  arc  of  the  horizon,   ES,  are 
evidently  divided  by  the  horizon  into  two  unequal  parts, 
the  smaller  of  which  in  each  case  lies  above  the  horizon. 
Hence  these  stars  are  above  the  horizon  less  than  twelve 
hours.     On   the   other   hand,    the   greater   portions  of  the 
circles  of  those  stars  which  rise  between  E  and  H  lie  above 
the  horizon,  and  the  stars  themselves  are  above  it  for  more 
than  twelve  hours. 

Any  body,  then,  whose  declination  is  of  the  same  name 
as  the  elevated  pole  will  be  above  the  horizon  more  than 
twelve  hours,  while  a  body  whose  declination  is  of  a  different 
name  from  that  of  the  elevated  pole  will  be  above  the  horizon 
less  than  twelve  hours.  And  further:  those  bodies  which 
are  in  north  declination,  in  other  words,  to  the  north  of  the 
celestial  equator,  will  rise  to  the  north  of  east  and  set  to 
the  north  of  west;  while  bodies  in  south  declination  will 
rise  and  set  to  the  south  of  the  east  and  the  west  point. 

When  a  star  has  no  declination,  that  is  to  say,  is  on  the 
celestial  equator,  it  will  rise  due  east  and  set  due  west,  and 
will  remain  twelve  hours  above  the  horizon. 

14.  Right   and   Parallel   Spheres. — When   an   observer 
travels  towards  the  elevated  pole,  the  radius  of  the  circle  of 
perpetual  apparition,   being  equal  to  the  altitude  of  the 
pole,  continually  increases,  and  the  number  of  stars  which 
never  set  increases  in  like  manner.     The  number  of  stars 
which  never  rise  also  increases.     Finally,  if  he  reaches  the 


SPHERICAL  CO-ORDINATES  17 

pole,  the  celestial  equator  coincides  with  the  horizon,  the 
east  and  the  west  point  disappear,  and  the  bodies  which  are 
on  the  same  side  of  the  equator  with  the  observer  are 
perpetually  above  the  horizon,  and  revolve  in  circles  whose 
planes  are  parallel  to  it,  while  the  bodies  which  are  on  the 
opposite  side  of  the  equator  never  rise.  As  he  travels 
towards  the  equator,  the  circles  of  perpetual  apparition  and 
occultation  alike  diminish,  the  diurnal  circles  become  more 
and  more  nearly  vertical,  and  when  he  reaches  the  equator, 
the  equinoctial  becomes  perpendicular  to  the  horizon  and 
coincides  with  the  prime  vertical,  and  the  horizon  bisects  all 
the  diurnal  circles.  At  the  equator,  then,  every  celestial 
body  comes  above  the  horizon,  and  remains  above  it  twelve 
hours. 

15.  Spherical  Co-ordinates. — The  position  of  any  point 
on  the  surface  of  a  sphere  is  determined,  as  soon  as  its  angular 
distances  are  given  from  any  two  great   circles  on  that 
sphere  whose  positions  are  known.     Thus  the  geographical 
position  of  any  point  on  the  earth's  surface  is  known  when  we 
have  determined  its  latitude  and  longitude;  in  other  words, 
when  we  know  its  angular  distance  from  the  equator  and 
from  the  prime  meridian.     In  like  manner  we  know  the 
position  of  any  point  on  the  surface  of  the  celestial  sphere 
when  either  its  altitude  and  azimuth,  or  its  right  ascension 
and  declination,  are  given.    In  the  first  of  these  two  systems 
of  co-ordinates  the  fixed  great  circles  are  the  celestial  horizon 
and  the  celestial  meridian,  the  origin  of  co-ordinates  being 
either  the  north  or  the  south  point  of  the  horizon.     In  the 
second  system  the  fixed  great  circles  are  the  equinoctial 
and  the  circle  of  declination  which  passes  through  the  vernal 
equinox,  and  the  origin  of  co-ordinates  is  the  vernal  equinox. 
In  Fig.  3,  if  we  know  the  arcs  HG  and  GA,  or  the  arcs  VB 
and  BA,  we  evidently  know  the  position  of  A. 

16.  Vanishing  Points  and  Vanishing  Circles. — Everyone 
knows  that  as  he  increases  the  distance  between  himself  and 
any  object,  the  apparent  magnitude  of  the  object  decreases; 


18     GENERAL  PHENOMENA  OF  THE  HEAVENS 

and  that,  if  he  recedes  far  enough  from  it,  it  will  be  reduced 
in  appearance  to  a  point.  Everyone  also  knows  that  when 
he  looks  along  the  line  of  a  railroad  track  the  lines  appear 
to  converge,  and  that,  if  the  track  is  straight,  and  the 
curvature  of  the  earth  does  not  limit  his  vision,  the  rails 
will  ultimately  appear  to  meet. 

These  familiar  illustrations  will  serve  to  show  what  is 
meant  by  a  vanishing  point.  The  actual  distance  between 
the  rails  of  course  remains  the  same;  but  the  angle  at  the 
eye  which  this  distance  subtends  decreases  as  the  eye  is 
directed  along  the  track,  until  at  last  it  ceases  to  subtend 
any  appreciable  angle  at  the  eye,  and  the  rails  apparently 
meet.  This  point  where  the  rails  appear  to  meet  is  called 
the  vanishing  point  of  the  two  lines;  and,  in  general,  the 
vanishing  point  of  any  system  of  parallel  lines  is  the  point 
at  which  they  will  appear  to  meet,  when  indefinitely  pro- 
longed. We  have  already  seen  (Art.  10)  that  the  pole  of 
the  heavens  is  the  vanishing  point  of  lines  drawn  per- 
pendicular to  the  equator,  and  the  same  may  be  said  of  the 
poles  of  any  circle  on  the  celestial  sphere.  For  instance,  the 
poles  of  the  horizon  at  any  place  are  the  zenith  and  the 
nadir;  and  any  system  of  lines  perpendicular  to  the  horizon 
will  apparently  meet,  when  prolonged  indefinitely,  in  these 
two  points.  And  again,  the  east  and  the  west  point  of  the 
horizon  are  the  poles  of  the  meridian,  and  lines  drawn  per- 
pendicular to  the  meridian  will  have  these  points  for  their 
vanishing  points. 

The  same  principle  holds  good  when  applied  to  any 
system  of  parallel  planes.  They  will  appear  to  meet,  when 
indefinitely  extended,  in  one  great  circle  of  the  sphere,  and 
this  circle  is  called  the  vanishing  circle  of  that  system  of 
planes.  The  celestial  horizon  is,  as  has  already  been  stated 
(Art.  4),  the  vanishing  circle  of  the  planes  of  the  sensible 
and  the  rational  horizon,  and  indeed  of  any  number  of 
planes  passed  parallel  to  them.  The  celestial  equator  is  the 
vanishing  circle  of  the  planes  of  all  the  parallels  of  latitude, 


SPHERICAL  PROJECTIONS  19 

and,  in  short,  every  circle  of  the  celestial  sphere  may  be 
regarded  as  the  vanishing  circle  of  a  system  of  planes  passed 
perpendicular  to  the  line  which  joins  the  poles  of  that  circle. 

17.  Spherical  Projections. — The  points  and  circles  of 
either  the  earth  or  the  celestial  sphere,  or  of  both,  may  be 
projected  upon  the  plane  of  any  great  circle  of  either  sphere. 
The  plane  on  which  the  projections  are  made  is  called  the 
primitive  plane,  and  the  circle  which  bounds  this  plane  is 
called  the  primitive  circle.  Several  distinct  methods  of  pro- 
jection will  be  found  in  treatises  on  Descriptive  Geometry. 
Of  these,  the  most  common  are  the  orthographic  and  the 
stereographic  projection. 

In  the  orthographic  projection,  the  point  of  sight  is  taken 
in  the  axis  of  the  primitive  circje,  and  at  an  infinite  distance 
from  that  circle.  All  circles  whose  planes  are  perpendicular 
to  the  primitive  plane  are  projected  into  right  lines;  all 
circles  whose  planes  are  parallel  to  the  primitive  plane  are 
projected  into  circles,  each  of  which  is  equal  to  the  circle 
of  which  it  is  a  projection;  and  all  other  circles  are  projected 
into  ellipses. 

In  the  stereographic  projection,  the  point  of  sight  is  at 
either  pole  of  the  primitive  circle,  and  its  distance  from  that 
circle  is  finite.  In  this  projection  every  circle  is  projected 
as  a  circle,  unless  its  plane  passes  through  the  point  of  sight, 
in  which  case  it  is  projected  into  a  right  line. 

For  the  construction  of  charts  representing  the  earth's 
surface  three  principal  projections  are  used.  M creator's 
projection,  in  which  the  earth's  surface  is  projected  on  the 
surface  of  a  cylinder  tangent  to  all  points  of  the  earth's 
equator,  represents  the  parallels  of  latitude  by  parallel  right 
lines  and  the  meridians  also  by  parallel  right  lines  per- 
pendicular to  the  parallels  of  latitude.  In  this  projection 
the  meridians  are  equidistant  but  the  distance  between 
successive  parallels  increases  as  we  recede  from  the  equator. 
The  advantage  offered  to  navigators  by  this  projection  is 
that  the  ship's  track,  as  long  as  the  course  on  which  it  sails 


20  GENERAL  PHENOMENA  OF  THE  HEAVENS 

is  unaltered,  is  represented  on  the  chart  by  a  straight  line, 
and  the  angle  which  this  line  makes  with  the  meridians 
is  the  course.  In  high  latitudes  there  is  excessive  dis- 
tortion on  charts  made  on  this  projection.  The  polyconic 
projection  is  the  development  of  the  earth's  surface  on  a 
series  of  cones  tangent  to  the  earth  at  successive  parallels 
of  latitude;  in  charts  constructed  on  this  projection  the 
parallels  of  latitude  and  the  meridians  are  both  developed  as 
curves.  There  is  less  distortion  than  on  a  Mercator  chart, 
but  it  is  less  convenient  for  navigators.  The  third  projection 
employed  for  chart  construction  is  the  gnomonic;  in  this 
the  earth's  surface  is  projected  by  rays  from  the  center  upon 
a  plane  tangent  to  the  surface  at  a  given  point;  all  great 
circles  are  projected  as  straight  lines,  hence  charts  con- 
structed on  the  gnomonic  projection  are  frequently  called 
great  circle  charts. 


CHAPTER  II 
ASTRONOMICAL  INSTRUMENTS.    ERRORS 

18.  THIS  chapter  will  be  devoted  to  a  general  description 
of  the  common  astronomical  instruments,  of  the  class  of 
observations  to  which  each  is  adapted,  and  of  the  manner  in 
which  such  observations  are  made.     No    attempt  will   be 
made  to  describe  the  elaborate  mechanism  by  which,  in 
many  cases,  the  usefulness  of  the  instrument  is  increased 
and  its  manipulation  is  facilitated;  but  enough,  it  is  hoped, 
will  be  said  to  enable  the  student  to  form  a  clear  conception 
of  the   prominent  features   of  each  instrument   which  is 
described.     There  is  no  lack  of  excellent  treatises  on  Astron- 
omy, in  which  those  who  wish  to  investigate  this  subject 
more  thoroughly  will  find  all  the  details,  which  the  limits 
prescribed  to  this  book  will  not  permit  to  enter  here,  clearly 
and  elaborately  presented, 

THE  ASTRONOMICAL  CLOCK 

19.  The  astronomical  clock  is  a  clock  which  is  regulated 
to  keep  sidereal  time,  and  is  an  indispensable  companion  to 
the  other  astronomical  instruments.     It  is  provided  with  a 
pendulum  so  constructed  that  change  of  temperature  will 
not  affect  its  length.     The  sidereal  day  at  any  place  com- 
mences, as  has  already  been  stated,  when  the  vernal  equinox 
is  on  the  upper  branch  of  the  meridian  of  that  place,  and  the 
theory  of  the  sidereal  clock  is  that  it  shows  Oh.  Om.  Os. 
when  the  vernal  equinox  is  so  situated.     Practically,  how- 
ever, it  is  found  that  every  clock  has  a  daily  rate;   that  is 

21 


22  ASTRONOMICAL  INSTRUMENTS,  ERRORS 

to  say,  it  gains  or  loses  a  certain  amount  of  time  daily.  In 
order,  then,  that  a  clock  may  be  regulated  to  sidereal  time, 
it  is  necessary  to  know  both  its  error  and  its  daily  rate; 
the  error  being  the  amount  by  which  it  is  fast  or  slow  at 
any  given  time,  and  the  daily  rate  being  the  amount  which 
it  gains  or  loses  daily;  and  knowing  these,  it  is  evidently  in 
our  power  to  obtain  at  any  desired  instant  the  true  sidereal 
time  from  the  time  shown  by  the  face  of  the  clock. 

It  is  to  be  noticed  further,  that,  except  as  a  matter  of 
convenience,  a  small  rate  has  no  advantage  over  a  large  one ; 
but  it  is  very  important .  that  the  rate,  whether  large  or 
small,  shall  be  constant  from  day  to  day;  so  that,  of  two 
clocks,  one  of  which  has  a  large  and  constant  rate,  and 
the  other  a  small  and  varying  one,  the  preference  is  to  be 
given  to  the  former. 

The  "standard"  pendulum  clock  of  an  observatory  is 
usually  kept  in  a  vault  or  airtight  case  where  the  temperature 
and  barometric  pressure  are  not  allowed  to  vary  appreciably, 
as  it  is  found  that  despite  extreme  care  in  manufacture  and 
in  operation,  the  rate  of  the  best  clock  will  be  affected  by 
changes  in  either. 

Clocks  may  be  set  to  keep  either  local  sidereal  time  or 
Greenwich  sidereal  time. 

20.  Error  of  the  Clock. — To  obtain  the  error  of  a  clock  on 
tne  local  sidereal  time  at  any  observatory,  we  make  use  of 
the  proposition,  already  demonstrated  (Art.  9),  that  the 
right  ascension  of  any  celestial  body,  when  at  its  upper 
culmination  on  any  meridian,  is  equal  to  the  sidereal  time 
at  that  meridian.  The  Ephemeris  gives  the  right  ascensions 
of  more  than  a  hundred  stars  which  are  suitable  for  observa- 
tions for  time.  By  means  of  an  instrument,  properly 
adjusted,  we  determine  the  instant  when  any  one  of  these 
stars  is  on  the  meridian,  and  the  time  which  the  clock  shows 
at  that  instant  is  noted.  This  is  the  clock  time  of  transit, 
and  the  right  ascension  of  the  star,  taken  from  the  Ephemeris, 
is  the  true  time  of  transit;  and  a  comparison  of  these  two 


THE  ASTRONOMICAL  CLOCK  23 

times  will  evidently  give  us  the  amount  by  which  the  clock 
is  fast  or  slow  on  local  sidereal  time. 

21.  Daily  Rate. — If,  in  a  similar  manner,  we  obtain  the 
error  of  the  clock  on  the  next  or  on  any  subsequent  day,  the 
difference  of  these  two  errors  will  be  the  gain  or  loss  of  the 
clock  in  the  interval;  and  hence,  if  we  divide  this  difference 
by  the  number  of  the  days  and  parts  of  days  which  have 
intervened  between  the  two  observations,  the  quotient  will 
be  the  daily  gain  or  loss. 

22.  Chronograph. — The  accuracy  of  astronomical  obser- 
vations is  much  enhanced  by  recording  the  times  of  the 
observations  by  means  of  an  electric  current.     A  cylinder 
about  which  a  roll  of  paper  is  wound,  is  turned  about  on 
its  axis  with  a  uniform  motion  by  the  use  of  appropriate 
machinery.     A  pen  is  pressed  down  upon  the  paper,  and  is 
so  connected  with  a  battery  that  whenever  the  circuit  is 
broken  a  mark  of  some  kind  is  made  upon  the  paper.     The 
wires  of  this  battery  are  connected  with  the  sidereal  clock 
in  such  a  way  that  every  oscillation  of  the  pendulum  breaks 
the  circuit.    Every  second  is  thus  recorded  upon  the  revolving 
paper.     The  observer  also  holds  in  his  hands  a  break-circuit 
key,  with  which,  whenever  he  wishes  to  note  the  time,  he 
breaks  the  circuit,  and  thus  causes  the  pen  to  make  its 
mark  upon  the  paper.     The  line  which  the  pen  describes  upon 
the  paper  will  be  something  like  this: 

i  I  I 

I I    i 1_J i        I 

abed 

The  equidistant  marks,  a,  6,  c,  etc.,  are  the  marks  caused  by 
the  pendulum,  and  the  marks  A,  B,  C  are  the  marks  which 
the  pen  makes  when  the  circuit  is  broken  by  the  observer. 
The  distance  from  A  to  6,  B  to  c,  etc.,  can  be  measured 
by  a  scale  of  equal  parts,  and  the  time  of  an  observation  can 
be  thus  obtained  within  a  small  fraction  of  a  second. 

The  cylinder  is  also  moved  by  a  screw  in  the  direction 
of  its  own  length,  so  that  the  pen  records  in  a  spiral. 


24  ASTRONOMICAL  INSTRUMENTS,  ERRORS 

This  instrument  is  called  a  Chronograph.  There  are 
several  varieties  of  the  chronograph  in  use  by  different 
astronomers,  but  the  main  principle  in  all  of  them  is  similar 
to  that  of  the  instrument  just  now  described. 

THE  ASTRONOMICAL  TELESCOPE 

23.  Telescopes  used  in  astronomical  work  are  of  two 
kinds,  refracting  and  reflecting.  In  both,  the  fundamental 
principle  is  the  same.  The  large  lens  of  the  refracting  type, 
or  the  mirror  of  the  reflecting  instrument,  forms  at  its  focus 
a  real  image  of  the  object  observed  and  this  image  is  then 
examined  and  magnified  by  the  lens  of  the  eye-piece  which 
in  principle  is  only  a  magnifying  glass. 

Theoretically,  the  refracting  telescope  has  only  two 
lenses:  the  large  lens  which  is  turned  toward  the  sky  is 
the  object  glass  and  the  smaller  lens  at  the  observer's  eye 
called  the  eye-piece.  A  single  lens  with  spherical  surfaces, 
cannot  gather  perfectly  to  a  single  point  all  the  rays  which 
enter  the  field  from  the  single  point  of  the  object  observed, 
the  "aberrations"  being  of  two  kinds,  spherical  and  chromatic. 
The  spherical  aberration  of  a  lens  results  in  the  rays  which 
pass  through  the  glass  near  its  edge  coming  to  a  shorter  focus 
than  those  which  pass  through  its  center,  while  the  chromatic 
aberration  is  the  separation  of  the  light  of  different  colors  due 
to  the  difference  in  refrangibility  of  the  rays  of  the  different 
colors.  The  chromatic  aberration  of  a  telescope  cannot  be 
corrected  by  any  modification  of  the  lens  itself,  but  it  has  been 
found  that  it  can  be  nearly  corrected  by  making  the  object- 
glass  of  two  (or  more)  lenses,  one  convex  of  crown  glass,  the 
other  concave  and  of  flint  glass.  At  the  same  time,  by  prop- 
erly choosing  the  curves  of  the  lenses,  the  spherical  aberration 
can  be  greatly  reduced.  A  Jens  of  this  kind  is  called  an 
achromatic  lens.  In  modern  instruments  both  object-glass  and 
eye-piece  are  compound  lenses;  each  consists  of  two  or  three 
separate  lenses,  placed  close  together  or  in  actual  contact. 


THE  TRANSIT  INSTRUMENT  25 

In  the  reflecting  telescope  the  place  of  the  object-glass 
is  taken  by  a  concave  mirror,  made  of  glass  and  silvered  on 
the  front  surface,  called  a  speculum.  By  using  another 
mirror  near  the  focus  of  the  speculum,  the  image  is  reflected 
to  the  focus  of  the  eye-piece  which  is  similar  to  that  used 
with  the  refractor.  The  great  advantage  of  the  reflector 
is  the  absence  of  all  chromatic  aberration  which  makes 
this  type  particularly  suitable  for  photographic  and  spec- 
troscopic  work.  It  is  possible  also  to  make  reflectors, 
larger  than  refractors;  the  great  Yerkes  telescope  has  an 
object  glass  forty  inches  in  diameter  while  a  new  reflector  at 
the  Mount  Wilson  (California)  Observatory  has  a  mirror 
100  inches  in  diameter.  For  apertures  up  to  20  inches  the 
refractor  transmits  more  light  than  the  reflector  and  gives 
better  definition  of  objects  over  a  larger  field.  The  well- 
equipped  observatory  has  telescopes  of  both  types,  as  each 
has  certain  advantages  over  the  other. 


THE  TRANSIT  INSTRUMENT 

24.  The  transit  instrument  is  used,  as  its  name  implies,  in 
observing  the  transits  of  the  heavenly  bodies.  Fig.  5 
represents  a  transit  instrument.  It  consists  of  a  telescope, 
TT,  sustained  by  an  axis,  A  A,  at  right  angles  to  it.  The 
extremities  of  this  axis  terminate  in  cylindrical  pivots, 
which  rest  in  metallic  supports,  VV,  shaped  like  the  upper 
part  of  the  letter  Y,  and  hence  called  the  Y;s.  These  Y's 
are  imbedded  in  two  stone  pillars.  In  order  to  relieve  the 
pivots  of  the  friction  to  which  the  weight  of  the  telescope 
subjects  them,  and  to  facilitate  the  motion  of  the  telescope,, 
there  are  two  counterpoises,  WW,  connected  with  levers, 
and  acting  at  X  X,  where  there  are  friction  rollers  upon 
which  the  axis  turns.  When  the  instrument  is  properly 
adjusted,  the  telescope  as  it  turns  about  with  the  axis  A  A, 
will  continually  lie  in  the  plane  of  the  meridian;  and,  in 


26  ASTRONOMICAL  INSTRUMENTS,  ERRORS 


FIG.  5. 


THE  TRANSIT  INSTRUMENT  27 

order  to  effect  this,  the  axis  A  A  should  point  to  the  east 
and  the  west  point  of  the  horizon  and  be  parallel  to  its 
plane.  There  are  therefore  screws  at  the  ends  of  the  axis, 
by  which  one  extremity  of  the  axis  may  be  raised  or  depressed, 
and  may  also  be  moved  forward  or  backward. 

The  Reticule. — In  the  common  focus  of  the  object-glass 
and  the  eye-piece  is  placed  the  reticule, 
a  representation  of  which  is  given 
in  Fig.  6.  It  consists  of  several  equi- 
distant vertical  wires  (usually  seven) 
and  two  horizontal  ones.  If  the  in- 
strument is  accurately  adjusted  to  the 


plane  of  the  meridian,  the  instant  that  FIQ  6 

any  star  is  on  the   middle  wire  is  the 

instant  of  its  transit.      These    wires  are   also   called  the 

cross-wires. 

25.  Adjustment. — The  axis  of  rotation  of  the  instrument 
is  an  imaginary  line  connecting  the  central  points  of  the 
pivots. 

The  axis  of  collimation  is  an  imaginary  line  drawn  from 
the  optical  center  of  the  object-glass,  perpendicular  to  the 
axis  of  rotation. 

The  line  of  sight  is  an  imaginary  line  drawn  from  the 
optical  center  of  the  object-glass  to  the  middle  wire. 

The  transit  instrument  is  accurately  adjusted  in  the  plane 
of  the  meridian,  when  the  line  of  sight  of  the  telescope  lies 
continually  in  that  plane,  as  the  telescope  revolves.  Three 
things,  then,  are  readily  seen  to  be  necessary:  the  axis  of 
rotation  must  be  exactly  horizontal;  it  must  lie  exactly  east 
and  west;  and  the  line  of  sight  and  the  axis  of  collimation 
must  exactly  coincide.  Practically,  these  conditions  are 
rarely  fulfilled;  but  they  can,  by  repeated  experiments,  be 
very  nearly  fulfilled,  and  the  errors,  which  the  failure  rigor- 
ously to  adjust  the  instrument  causes  in  the  observations, 
will  be  constant  and  small,  and  can  be  accurately  deter- 
mined. 


28  ASTRONOMICAL  INSTRUMENTS,  ERRORS 

26.  Application. — The  principal  application  of  the  transit 
instrument  in  observatories  is  to  the  determination  of  the 
right  ascensions  of  celestial  bodies.     Knowing  the  constant 
instrumental  errors  just  now  mentioned,  and  the  error  and 
the  rate  of  the  clock,  we  can  easily  obtain  the  true  sidereal 
time  at  the  instant  of  the  transit  of  a  celestial  body,  which 
time  is  at  once,  as  we  have  already  seen,  the  right  ascension 
of  that  body. 

THE  MERIDIAN  CIRCLE 

27.  The  meridian  circle  is  a  combination  of  a  transit 
instrument,    similar    to    the    one  above    described,  and    a 
graduated  circle,  securely  fastened  at  right  angles  to  the 
horizontal  axis,  and  turning  with  it.     A  meridian  circle  is 
represented  in  Fig.  7.     The  horizontal  axis  bears  two  grad- 
uated circles,  CC,  C'C',  the  first  of  these  circles  being  much 
more  finely  graduated  than  the  second,  the  latter  being  only 
used  as  a  finder,  to  set  the  telescope  approximately  at  any 
desired  altitude.     R  and  R  represent  four  of  eight  stationary 
microscopes,  by  which  the  circles  CC  and  C'C'  are  read; 
LL  is  a  hanging  level,  by  which  the  horizontality  of  the  axis 
is  tested.     The  cross-wires  are  illuminated  by  light  which 
passes  from  a  lamp  through  the  tubes  A  A,  and  through 
the  pivots  which  are  perforated  for  this  purpose,  and  is 
reflected  towards  the  reticule  by  a  metallic  speculum  which 
is  set  within  the  hollow  cube  M.     The  quantity  of  light 
admitted  is  regulated   by   revolving   disks  with   eccentric 
apertures,  which  are  placed  between  the  Y's  and  the  tubes 
A  A,  and  are  moved  by  cords  carrying  small  weights,  SS. 

The  object  in  having  so  many  reading  microscopes  is  to 
diminish  the  errors  arising  from  the  imperfect  graduation 
of  the  vertical  circle.  When  any  angle  is  to  be  measured, 
the  readings  of  all  eight  microscopes  are  first  taken.  The 
telescope,  carrying  the  circle  with  it,  is  then  moved  through 
the  angle  whose  value  is  required,  and  the  new  readings  of 


THE  MERIDIAN  CIRCLE 


29 


the  microscopes,  which  have  remained  stationary,  are  taken. 
We  thus  obtain  eight  values,  one  from  each  microscope,  of 
•the  angle  measured.  Theoretically,  these  values  should  be 


FIG.  7. 


identical,  and  if  they  are  not,  their  mean  is  taken  as  the 
true  measure  of  the  angle  observed. 

28.  The  Reading  Microscope. — The  reading  microscope 
is  represented  in  Fig.  8.     The  observer,  placing  his  eye  at 


30 


ASTRONOMICAL  INSTRUMENTS,  ERRORS 


A,  sees  the  image  of  the  divisions  of  the  graduated  circle 

MN,  formed  at  D,  the  common  focus  of  the  glasses  A  and  C. 

He  will  also  see  a  scale  of 
notches,  nn,  and  two  in- 
tersecting spider  threads, 
as  shown  in  Fig.  9.  These 
threads  are  attached  to 
a  sliding  frame,  aa,  which 
is  moved  by  means  of  a 
fine  screw,  cc,  the  head  of 
which,  EF,  is  graduated. 
The  scale  of  notches 
is  immovable,  and  is 
so  constructed  that  the 

M  ^ N distance  between  the  cen- 

'  .  ters  of  any  two  consecu- 

FIG.  8  tive  notches  is  equal  to 

that  between  the  threads 

of  the  screw,  thus  making  the  number  of  teeth  passed  over 

by  the  spider  threads  equal  to  the  number  of   complete 

revolutions  made  by  the  screw.     The  central  notch  is  taken 

as  the  point  of  reference, 

and  is  distinguished  by 

a    hole    opposite    to    it. 

There  is  a  fixed  index  at 

t,  to  which  the  divisions 

on  the  head  of  the  screw 

are  referred.     When  any 

division  of  the  limb  does 

not    coincide    with     the 

central  notch,  the  spider  FIG.  9. 

threads  are  moved  from 

the  central   notch  to   the   division,   and    the    number    of 

revolutions    and    fractional    parts    of   a    revolution  which 

the  screw  makes  is  noted.     If  now  we  suppose  the  value 

of  each  division  of  the  graduated  circle,   MN,  to  be  10', 


THE  MERIDIAN  CIRCLE  31 

and  that  ten  revolutions  of  the  screw  suffice  to  carry  the 
spider  threads  across  one  of  these  divisions,  then  will  one 
revolution  of  the  screw  correspond  to  an  arc  of  1':  and 
if  we  further  suppose  that  the  head  of  the  screw  is  divided 
into  60  equal  parts,  then  each  division  on  the  head  will 
correspond  to  an  arc  of  1".  In  such  a  case,  the  com- 
plete reading  of  the  limb  is  obtained  to  the  nearest  second. 
By  increasing  the  power  of  the  microscope,  the  fineness  of 
the  screw,  and  the  number  of  the  graduations  on  the  screw- 
head,  the  reading  of  the  limb  may  be  obtained  with  far 
greater  precision. 

29.  Fixed   Points. — The  meridian   circle,   being  also   a 
transit  instrument,  may  be  used  as  such;  but  the  object  for 
which  it  is  especially  used  is  the  measurement  of  arcs  of  the 
meridian.     In  order  to  facilitate  such  measurement,  certain 
fixed  points  of  reference  are  determined  upon  the  vertical 
circle.     The  most  important  of  these  points  are  the  horizontal 
point,  by  which  is  meant  the  reading  of  the  instrument  when 
the  axis  of  the  telescope  lies  in  the  plane  of  the  horizon; 
the  polar  point,  which  is  the  reading  of  the  instrument  when 
the  telescope  is  directed  to  the  elevated  pole;    the  zenith 
point,  and  the  nadir  point. 

30.  The  Horizontal  Point. — As  the  surface  of  a  fluid, 
when  at  rest,  is  necessarily  horizontal,  and  as,  by  the  laws 
of  Optics,  the  angles  of  incidence  and  reflection  are  equal  to 
each  other,  the  image  of  a  star  reflected  in  a  basin  of  mercury 
will  be  depressed  below  the  horizon  by  an  angle  equal  to  the 
altitude  of  the  star  at  that  instant.     If,  then,  we  take  the 
reading  of  the  vertical  circle  when  a  star  which  is  about  to 
cross  the  meridian  is  on  the  first  vertical  thread  of  the  reticule, 
and  then,  depressing  the  telescope,  take  the  reading  of  the 
circle  when  the  reflected  image  of  the  star  crosses  the  last 
vertical  thread,  and,  by  means  of  small  corrections,  reduce 
these  readings  to  what  they  would  have  been,  had  both  star 
and  image  been  on  the  meridian  when  the  observations  were 
made,   the  mean  of  these  two  reduced  readings  will  be 


32 


ASTRONOMICAL  INSTRUMENTS,  ERRORS 


the  horizontal  point.  The  horizontal  point  having  been 
thus  determined,  the  zenith  point  and  the  nadir  point,  being 
situated  at  intervals  of  90°  from  it,  are  at  once  obtained. 
Knowing  the  horizontal  and  the  zenith  point,  we  are  able  to 
measure  the  meridian  altitude  or  the  meridian  zenith  distance 
of  any  celestial  body  which  comes  above  our  horizon.  And 
further,  as  the  latitude  is  equal  to  the  altitude  of  the  elevated 
pole,  if  the  latitude  of  the  place  is  accurately  known,  we  can 
at  once  obtain  the  polar  point  by  applying  the  latitude  to 
the  horizontal  point. 

31.  Nadir  Point. — The  nadir  point  may  be  independently 

obtained  in  the  following  manner:  Let 
the  telescope,  represented  in  Fig.  10  by 
A  B,  be  directed  vertically  downwards 
towards  a  basin  of  mercury,  CD.  The 
observer,  placing  his  eye  at  A,  will  see 
the  cross-wires  of  the  telescope,  and  will 
see  also  the  image  of  these  wires  reflect- 
ed into  the  telescope  from  the  mercury. 
By  slowly  moving  the  telescope,  the 
cross-wires  and  their  reflected  image  may 
be  brought  into  exact  coincidence,  and 
the  reflected  image  will  then  disappear. 
The  line  of  sight  of  the  telescope  is  now 
vertical,  and  the  reading  of  the  vertical 
circle  will  be  the  nadir  point,  from  which  the  other  points 
can  readily  be  found. 

There  is  a  variety  of  methods  by  which  each  of  these 
points  can  be  obtained,  without  reference  to  any  other;  and 
by  comparing  the  results  which  these  different  independent 
methods  give,  the  errors  to  which  each  result  is  liable  may 
be  very  considerably  diminished. 

32.  Use  of  the  Meridian  Circle. — The  meridian  circle 
may  be  used  in  connection  with  the  sidereal  clock,  to  find 
the  right  ascension  and  declination  of  any  celestial  body. 
The  telescope  is  directed  towards  the  body  as  it  crosses  the 


FIG.  10. 


THE  ALTITUDE  AND  ASIMUTH  INSTRUMENT        33 


meridian,  and  the  time  of  transit  as  shown  by  the  clock, 
and  the  reading  of  the  vertical  circle,  are  both  taken.  We 
have  already  seen  how  the  right  ascension  of  the  body  is 
obtained  frqm  the  clock  time  of  transit.  The  difference 
between  the  reading  of  the  circle,  which  we  suppose  to  have 
been  taken,  and  the  polar  point,  is  the  polar  distance  of 
the  body,  the  complement  of  which  is  the  declination. 
Or  we  may  obtain  the  declination  still  more  directly  by 
previously  establishing  the  equinoctial  point  of  the  instru- 
ment, the  reading,  that  is  to  say,  of  the  vertical  circle  when 
the  telescope  lies  in  the  plane  of  the  equinoctial.* 

On  the  other  hand,  if  we  make  the  same  observations 
upon  a  star  whose  right  ascension  and  declination  are  known, 
we  can  determine  the  latitude  and  the  sidereal  time  of  the 
place  of  observation. 

THE  ALTITUDE  AND  AZIMUTH  INSTRUMENT 

33.  The  general  principles  on  which  the  altitude  and 
azimuth  instrument  is  constructed 
are  seen  in  Fig.  11.  Through 
the  center  of  a  graduated  circle, 
C'C'j  and  perpendicular  to  its 
plane,  is  passed  an  axis,  A  A. 
At  right  angles  to  this  axis  is  a 
second  axis,  one  extremity  of 
which  is  represented  by  B.  This 
second  axis  carries  the  telescope, 
TT,  and  also  a  second  graduated 
circle,  CC,  whose  plane  is  per- 
pendicular to  that  of  the  circle 
C'C'.  The  telescope  admits  of  being  moved  in  the  plane 

*  As  the  direction  in  which  a  star  appears  to  lie  is  not,  owing  to  re- 
fraction and  other  causes,  which  will  be  explained  in  Chap.  III.,  the 
direction  in  which  it  really  lies,  certain  small  corrections  must  be  ap- 
plied to  the  reading  of  the  circle  to  obtain  the  reading  which  really 
corresponds  to  the  direction  of  the  star. 


FIG.  11. 


34  ASTRONOMICAL  INSTRUMENTS,  ERRORS 

of  each  circle,  and  microscopes  or  verniers  are  attached 
to  the  instrument,  by  means  of  which  arcs  on  either  circle 
can  be  read. 

If  this  instrument  is  so  placed  that  the  principal  axis, 
A  A,  lies  in  a  vertical  direction,  we  shall  have  an  altitude 
and  azimuth  instrument,  sometimes  called  an  altazimuth. 
The  circle  C'C'  will  then  lie  in  the  plane  of  the  horizon,  and 
the  axis  A  A,  indefinitely  prolonged,  will  meet  the  surface 
of  the  celestial  sphere  in  the  zenith  and  the  nadir.  The 
cirple  CC,  as  it  is  moved  with  the  telescope  about  the  axis 
A  A,  will  continually  lie  in  a  vertical  plane. 

34.  Fixed  Points. — Altitudes  may  be  measured  on  the 
vertical  circle,  when  we  know  the  horizontal  or  the  zenith 
point,  the  determination  of  which  has  already  been  described 
in  Arts.  30  and  31.     In  order  to  measure  the  azimuth  of  any 
celestial  body,  we  must,  in  like  manner,  establish  some  fixed 
point  of  reference  on  the  horizontal  circle,  as,  for  instance, 
the  north  or  the  south  point,  by  which  is  meant  the  reading 
of  the  horizontal  circle  when  the  telescope  lies  in  the  plane 
of  the  meridian. 

35.  Method   of   Equal   Altitudes. — One  of  the  methods 

of  obtaining  the  north  or  the  south 
point  of  the  horizontal  circle  is 
called  the  method  of  equal  altitudes. 
Let  Fig.  12  represent  the  projection 
of  the  celestial  sphere  on  the  plane 
of  the  celestial  horizon,  NESW. 
Z  is  the  projection  of  the  zenith, 
P  of  the  pole,  and  the  arc  A  A' 
the  projection  of  a  portion  of  the 
FIG.  12.  diurnal  circle  of  a  fixed  star,  which 

is    supposed    to    have    the    same 

altitude  when  it  reaches  A',  west  of  the  meridian,  which  it 

had  at  A,  east  of  the  meridian. 

Now,  in  the  two  triangles  PAZ,  PA'Z,  we  have  PZ 

common,  PA'  equal  to  PA  (since  the  polar  distance  of  a 


THE  ALTITUDE  AND  ASIMUTH  INSTRUMENT         35 

fixed  star  remains  constant),  and  ZA  equal  to  ZA',  by 
hypothesis;  the  two  triangles  are  therefore  equal  in  all  their 
parts,  and  hence  the  angles  PZA  and  PZA'  are  equal. 
But  these  two  Bangles  are  the  azimuths  of  the  star  at  the 
two  positions  A  and  A' '.  We  may  say,  then,  in  general, 
that  equal  altitudes  of  a  fixed  star  correspond  to  equal 
angular  distances  from  the  meridian. 

Now,  let  the  telescope  be  directed  to  some  fixed  star 
east  of  the  meridian,  and  let  the  reading  of  the  horizontal 
circle  be  taken.  When  the  star  is  at  the  same  altitude, 
west  of  the  meridian,  let  the  reading  of  the  horizontal  circle 
again  be  taken;  the  mean  of  these  two  readings  is  the 
reading  of  the  horizontal  circle  when  the  axis  of  the  tele- 
scope lies  in  the  plane  of  the  meridian. 

36.  Use  of  the  Altitude  and  Azimuth  Instrument. — This 
instrument  is  chiefly  used  for  the  determination  of  the 
amount  of  refraction  corresponding  to  different  altitudes. 
Refraction,  as  will  be  seen  in  the  next  chapter,  displaces 
every  celestial  body  in  a  vertical  direction,  making  its 
apparent  zenith  distance  less  than  its  true  zenith  distance. 
At  the  instant  of  taking  the  altitude  of  a  celestial  body, 
the  local  sidereal  time  is  noted,  from  which,  knowing  the 
right  ascension  of  the  body,  we  can  obtain  its  hour  angle 
from  the  theorem  in  Art.  9.  We  shall  then  have  in  the 
astronomical  triangle,  PZA  (Fig.  3),  the  hour  angle  ZPA, 
the  side  PA,  or  the  polar  distance  of  the  body,  and  the  side 
PZ,  the  co-latitude  of  the  place  of  observation.  We  can, 
therefore,  compute  the  side  ZA,  which  is  the  true  zenith 
distance  of  the  body  observed;  and  the  difference  between 
this  and  the  observed  zenith  distance  (corrected  for  parallax, 
[Art.  55],  and  instrumental  errors),  will  be  the  amount  of 
refraction  for  that  zenith  distance. 

The  construction  of  the  instrument  enables  us  to  follow 
a  celestial  body  through  its  Avhole  course  from  rising  to 
setting,  measuring  altitudes  and  noting  the  corresponding 
times  to  any  extent  that  we  choose;  and  the  amount  of 


36  ASTRONOMICAL  INSTRUMENTS,  ERRORS 

refraction  corresponding  to  each  altitude  can  afterwards  be 
computed  at  our  leisure. 

THE  EQUATORIAL 

37.  The  equatorial  is  similar  in  general  construction  to 
the    altitude    and    azimuth    instrument.     It    is,    however, 
differently  placed,   the   plane   of  the   principal  graduated 
circle,  C'C",  coinciding,  not  with  the  plane  of  the  horizon, 
but  with  that  of  the  celestial  equator,  from  which  peculiarity 
of  position  comes  the  name  of  equatorial.    The  circle  C'C', 
when  thus  placed,  is  called  the  hour  circle  of  the  instrument, 
and  the  axis  A,  at  right  angles  to  it,  is  called  the  hour  or  polar 
axis.     It  is  evident  that  the  axis  is  directed  towards  the 
poles  of  the  heavens.     The  circle  CC,  the  plane  of  which  is 
perpendicular  to  the  plane  of  the  circle  C'C',  will  lie  con- 
tinually in  the  plane  of  a  circle  of  declination,  as  the  instru- 
ment is  turned  about  the  polar  axis,  and  is  hence  called  the 
declination  circle.    The  axis  on  which  this  latter  circle  is 
mounted  is  called  the  declination  axis. 

38.  Use  of  the  Equatorial. — The  equatorial  is  employed 
principally  in  that  class  of  observations  which  requires  a 
celestial  body  to  remain  in  the  field  of  view  during  a  con- 
siderable length  of  time.     The 
manner    in  which  this  require- 
ment  is    met    is    explained    in 
Fig.  13.     Let  A  A'  be  the  polar 
axis  of  an  equatorial,   directed 
towards  the  pole  of  the  heavens, 

>   P.      Let   ss's"   be   the    diurnal 
circle   in  which  a  star  appears 
FIG.  13.  to  move  about  the  pole.     Sup- 

pose  the   telescope,  TT,  to  be 

turned  in  the  direction  of  the  star  when  at  s,  and  to  be 
moved  until  the  intersection  of  the  cross-wires  and  the 
star  coincide,  and  then  clamped.  Now,  if  the  instru- 


THE  EQUATORIAL  37 

ment  is  made  to  revolve  about  the  axis  A  A',  with  an 
angular  velocity  equal  to  that  of  the  star  about  the  pole, 
it  is  plain  that,  since  the  angle  which  the  axis  of  the  telescope 
makes  with  the  polar  axis  remains  unchanged,  and  is  con- 
tinually equal  to  the  angular  distance  of  the  star  from  the 
pole,  the  coincidence  of  the  cross-wires  and  the  star  will 
remain  complete. 

A  clock-work  arrangement,  called  a  driving  clock,  is 
connected  with  most  equatorials,  by  which  the  telescope 
may  be  moved  uniformly  about  its  polar  axis  at  the  same 
rate  as  the  apparent  rotation  of  the  celestial  bodies  about 
the  celestial  pole.  This  rotation  of  the  telescope,  mounted 
equatorially,  is  necessary  in  order  to  keep  the  celestial  body 
within  the  field  of  the  telescope;  measurements  of  angular 
diameters  of  planets,  angular  distance  between  two  stars 
which  are  near  each  other,  and  other  micrometric  observa- 
tions of  a  similar  character  are  thus  made  possible.  In 
stellar  photography  it  is  essential  that  the  telescope  follow 
the  star  exactly;  to  ensure  this,  telescopes  used  for  this  pur- 
pose are  usually  fitted  with  a  visual  telescope  rigidly  secured 
parallel  to  the  photographic  telescope;  the  observer  at  the 
visual  telescope  can  adjust  the  clock  mechanism  as  necessary 
to  keep  the  photographic  telescope  continuously  pointed  at 
the  desired  point  in  the  heavens. 

The  micrometer  referred  to  above  is  an  instrument  used 
for  the  measurement  of  small  angles.  It  is  placed  in  the 
focus  of  the  telescope  and  the  principle  of  its  construction 
is  like  that  of  the  reading  microscope.  Its  essential  feature 
consists  of  two  parallel  wires  or  spider  lines,  both  movable, 
or  one  fixed  and  the  other  movable,  the  motion  being 
governed  by  screws,  as  in  the  reading  microscope.  The 
motion  of  the  wire  through  0.0001  inch  can  be  noted  by  the 
screw;  whence  it  is  evident  that  very  minute  angular  dis- 
tances can  be  measured. 


38  ASTRONOMICAL  INSTRUMENTS,  ERRORS 

THE  SPECTROSCOPE 

39.  If  light  passes  through  a  prism,  the  light  is  dispersed, 
or  spread  out,  due  to  the  different  refrangibilities  of  the 
light  waves  of  the  different  wave  lengths;  a  distant  point  of 
light,  as  a  star,  viewed  through  a  prism  will  appear  as  a  long 
streak  of  light,  red  at  one  end  and  violet  at  the  other.  If 
the  object  looked  at  be  not  a  point  but  a  line  of  light  parallel 
to  the  edge  of  the  prism,  then,  instead  of  a  mere  colored 
streak  without  width,  the  observer  will  get  a  colored  band 
of  light,  called  a  spectrum,  from  which  valuable  information 
may  be  gained. 

The  spectroscope  consists  of  a  tube  called  the  collimator 
having  at  one  end  a  narrow  slit,  S,  and  at  the  other  end  an 
achromatic  lens,  L,  at  such  a  distance  that  the  slit  is  in  its 
principal  focus.  The  light  rays  passing  through  the  colli- 
mator enter  the  prism  and  the  rays  are  dispersed.  The 
view  telescope  magnifies  the  spectrum.  Frequently,  the 
prism  is  replaced  by  a  diffraction  grating,  which  is  a  metal  or 
glass  plate  ruled  with  many  thousand  straight  equidistant 
lines.  For  photographing  stellar  spectra  a  plate  holder  is 
substituted  for  the  view  telescope.  In  some  spectroscopic 
observations,  three  prisms  are  used  instead  of  one  and  the 
direction  of  the  rays  is  changed  180°  with  greater  consequent 
dispersion.  There  is  also  an  arrangement,  not  shown  in 
the  diagram,  by  which  rays  of  light  from  two  bodies  or 
substances  can  be  introduced  without  interfering  with  each 
other,  so  that  their  spectra  can  be  formed  simultaneously, 
one  above  the  other,  and  the  points  of  resemblance  or  differ- 
ence between  them  can  be  accurately  noted. 

If  the  light  comes  from  an  incandescent  gas  at  low  pres- 
sure, the  spectrum  is  not  continuous,  but  is  made  up  of  a 
series  of  lines,  or  images  of  the  slit,  variously  located 
throughout  the  spectrum  and  the  combination  of  the  lines 
is  different  for  every  gas.  It  has  been  found  that  the  solar 
spectrum  contains  a  large  number  of  dark'  and  narrow 


THE  SPECTROSCOPE  39 

lines  and  that  the  spectra  of  stars  also  contain  similar  series 
of  lines,  differing  from  each  other;  each  series  is,  however, 
constant  for  the  same  body  under  the  same  conditions 
(the  spectrum  of  an  individual  star  may  change).  Hence 
by  comparison  of  the  spectra  of  the  heavenly  bodies  with 
those  of  known  chemical  substances,  the  existence  of  many 
of  those  substances  in  the  heavenly  bodies  has  been  defi- 
nitely established.  Nor  is  this  all,  the  inspection  of  any 
spectrum  suffices  to  tell  us  whether  the  light  which  forms 
it  comes  from  a  solid  or  a  gaseous  body,  and  whether,  if 
the  light  comes  from  a  solid  body,  it  passes  through  a 
gaseous  body  before  it  reaches  us,  and  by  micrometric 


View 
Telescope 


measurement  of  the  displacement  of  lines  in  the  spectrum 
it  is  possible  to  determine  the  velocity  with  which  a  star  is 
approaching  or  receding  from  us.  The  results  of  these 
investigations  will  be  noticed  when  we  come  to  the  de- 
scription of  the  heavenly  bodies;  and  the  method  of 
investigation  will  be  further  illustrated  in  the  Article  on 
the  composition  of  the  sun  (Art.  102). 

THE  SEXTANT 

40.  The  sextant  is  an  instrument  by  which  the  angular 
distance  between  two  visible  objects  may  be  measured.  It 
is  used  chiefly  by  navigators;  but  its  portability  gives  it 


40 


ASTRONOMICAL  INSTRUMENTS,  ERRORS 


great  value  wherever  celestial  observations  are  required. 
The  angles  for  the  measurement  of  which  it  is  used  are  the 
altitudes  of  celestial  bodies,  and  the  angular  distances 
between  celestial  bodies  or  terrestrial  objects. 

Fig.  14  is  a  representation  of  the  sextant.  Its  form  is 
that  of  a  sector  of  a  circle,  the  arc  of  which  comprises  75°. 
A  movable  arm,  CD,  called  the  index-bar,  revolves  about 
the  center  of  the  sector.  This  bar  carries  at  one  extremity 


a  vernier,  D.  At  the  other  extremity  of  the  index-bar,  and 
revolving  with  it,  is  placed  a  silvered  mirror,  C,  the  surface 
of  which  must  be  perpendicular  to  the  plane  of  the  instru- 
ment. This  glass  is  called  the  index-glass.  Another  glass, 
Nj  called  the  horizon-glass,  is  attached  to  the  frame  of  the 
instrument,  and  only  its  lower  half  is  silvered.  This  glass 
is  immovable,  and  its  surface  must  be  perpendicular  to  the 


THE  SEXTANT  41 

plane  of  the  instrument.  T  is  a  telescope,  directed  towards 
the  horizon-glass,  with  its  line  of  sight  parallel  to  the  plane 
of  the  instrument.  F  and  E  are  two  sets  of  colored  glasses, 
which  may  be  used  to  protect  the  eye  when  the  sun  is 
observed.  M  is  a  magnifying  glass,  to  assist  the  eye  in 
reading  the  vernier.  G  is  a  tangent  screw,  which  gives  a 
slight  motion  to  the  index-bar,  and  is  used  in  obtaining  an 
accurate  coincidence  of  the  images.  A  clamp  screw,  not 
shown,  secures  the  index  bar  to  the  arc. 

41.  Optical  Principle  of  Construction. — The  sextant  is 
constructed  upon  a  principle  in  Optics  which  may  be  stated 
thus:  The  angle  between  the  first  and  the  last  direction  of  a 
ray  which  has  suffered  two  reflections  in  the  same  plane  is 
equal  to  twice  the  angle  which  the  two  reflecting  surfaces  make 
with  each  other. 

To  prove  this:  In  Fig.  15,  let  A  and  B  be  the  two  re- 
flecting surfaces,  supposed  to  be  placed  with  their  planes 
perpendicular  to  the  plane  of  the  paper.  Let  SA  be  a  ray 
of  light  from  some  body,  S,  which  is  reflected  from  A  to 
B,  and  from  B  in  the 
direction  BE.  Prolong 
SA  until  it  meets  the 
line  BE.  Then  will  the 
angle  SEE  be  the  angle 
between  the  first  and 
the  last  direction  of  the 
ray  SA.  At  the  points 
A  and  B  let  the  lines 
AD  and  BC  be  drawn 
perpendicular  to  the  FIG.  15. 

reflecting   surfaces,    and 

prolong  AD  until  it  meets  BC.  The  angle  DCBjs  equal 
to  the  angle  which  the  two  surfaces  make  with  each 
other.  We  have  then  to  prove  that  the  angle  SEB  is 
double  the  angle  DCS. 

Now  since  the  angle  of  incidence  always  equals  the  angle 


42 


ASTRONOMICAL  INSTRUMENTS,  ERRORS 


of  reflection,  SAD  and  DAB  are  equal,  and  so  are  ABC 
and  CBE.  We  have,  by  Geometry, 

SEB=SAB-ABE 
=2(DAB-ABC), 
=  2DCB. 

42.  Measurement  of  Angular  Distances. — Suppose,  now, 
that  we  wish  to  measure  the  angular  distance  between  two 
celestial  bodies,  A  and  B  (Fig.  16).  The  instrument  is  so 
held  that  its  plane  passes  through  the  two  bodies,  and  the 
fainter  of  them,  which  in  this  case  we  suppose  to  be  B, 
is  seen  directly  through  the  horizon-glass  and  the  telescope. 
B  is  so  distant  that  the  rays  B'C  and  Bm,  coming  from  it, 

may  be  considered  to  be  sen- 
sibly parallel.  Let  db  and  CI 
be  the  positions  of  the  index- 
glass  and  index-bar  when  the 
index-glass  and  the  horizon- 
glass  are  parallel.  Then  will 
the  ray  B'C  be  reflected  by 
the  two  glasses  in  a  direction 
parallel  to  itself,  and  the  ob- 
server, whose  eye  is  at  D,  will 
see  both  the  direct  and  the 
reflected  image  of  B  in  coin- 
cidence. Now  let  the  index- 
bar  be  moved  to  some  new 

position,  CI',  so  that  the  ray  from  the  second  body,  A, 
shall  be  finally  reflected  in  the  direction  of  mD.  The 
observer  will  then  see  the  direct  image  of  B  and  the  reflected 
image  of  A  in  coincidence;  and  the  angular  distance  between 
the  two  bodies  is  evidently  equal  to  the  angle  between  the 
first  and  the  last  direction  of  the  ray  AC,  which  angle  has 
already  been  shown  to  be  equal  to  twice  the  angle  which  the 
two  glasses  now  make  with  each  other,  or  to  twice  the  angle 


FIG.  16. 


THE  SEXTANT  43 

1C  I'.  If,  then,  we  know  the  point  /  on  the  graduated  arc 
at  which  the  index-bar  stands  when  the  glasses  are  parallel, 
twice  the  difference  between  the  reading  of  that  point  and 
that  of  the  point  I'  will  be  the  angular  distance  of  the  two 
bodies. 

To  avoid  this  doubling  of  the  angle,  every  half  degree  of 
the  arc  is  marked  as  a  whole  degree,  when  the  graduation  is 
made;  so  that,  in  practice,  we  have  only  to  subtract  the 
reading  of  I  from  that  of  /'  to  obtain  the  angle  required. 

43.  Index  Correction. — The  point  of  reference  on  the  arc 
from  which  all  angles  are  to  be  reckoned  is,  as  we  have 
already  seen,  the  reading  of  the  sextant  when  the  surfaces 
of  the  index-glass  and  the  horizon-glass  are  parallel.  This 
point  may  fall  either  at  the  zero  of  the  graduation,  or  to 
the  left  or  to  the  right  of  it;  and  to  provide  for  the  last 
case,  the  graduation  is  carried  a  short  distance  to  the  right 
of  the  zero,  this  portion  of  the  arc  being  called  the  extra  arc. 
The  reading  of  this  point  of  parallelism  is  called  the  index 
error  and  the  correction  to  be  applied  to  correct  this  error 
is  the  index  correction.  This  index  correction  is  positive 
when  it  falls  to  the  right  of  the  zero  and  negative  when  it 
falls  to  the  left.  Suppose,  for  instance,  that  the  instru- 
ment reads  2'  on  the  extra  arc  when  the  glasses  are  parallel: 
all  angles  ought  then  to  be  reckoned  from  the  point  2', 
instead  of  from  the  zero  point;  in  other  words,  2'  is  a  con- 
stant correction  to  be  added  to  every  reading. 

There  are  several  methods  of  finding  the  index  correction. 
One  method,  which  can  readily  be  shown  from  Fig.  16  to  be 
a  legitimate  one,  is  to  move  the  index-bar  until  the  direct 
and  the  reflected  image  of  the  same  star  are  in  coincidence, 
and  then  take  the  reading,  giving  it  its  proper  sign  according 
to  the  rule  above  stated.  Other  methods,  generally  more 
convenient,  in  which  either  the  sun  or  the  horizon  is  used, 
may  be  found  in  Bowditch's  Navigation  and  in  most  treatises 
on  astronomy;  where  also  may  be  found  the  methods  of 
testing  the  adjustments  of  the  sextant. 


44  ASTRONOMICAL  INSTRUMENTS,  ERRORS 

44.  The    Artificial   Horizon. — In    order   to    obtain    the 
altitude  of  a  celestial  body  at  sea,  the  sextant  is  held  in  a 
vertical  position,  and  the  index-bar  is  moved  until  the  re- 
flected image  of  the  body  is  brought  into  contact  with  the 
visible  horizon  seen  through  the  telescope  of  the  sextant. 
The  sextant  reading  is  then  corrected  for  the  index  correction; 
and  corrections  must  also  be  applied  for  parallax,  refraction, 
and  the  dip  of  the  horizon,  as  will  be  explained  in  the  next 
chapter.     If  the  body  observed  is  the  sun  or  the  moon, 
either  its  upper  or  its  lower  limb  is  brought  into  contact 
with  the  horizon,  and  the  value  of  its  angular  semi-diameter 
(given  in  the  Nautical  Almanac)  is  subtracted  or  added. 

On  shore,  use  is  made  of  the  artificial  horizon,  already 
alluded  to  in  Art.  30.  This  commonly  consists  of  a  shallow, 
rectangular  basin  of  mercury,  the  surface  of  which  is  pro- 
tected from  the  wind  by  a  sloping  roof  of  glass.  The  observer 
so  places  himself  that  he  can  see  the  image  of  the  body 
whose  altitude  he  wishes  to  measure  reflected  in  the  mercury. 
He  then  moves  the  index-bar  of  the  sextant  until  the  image 
of  the  body  reflected  by  the  sextant  is  in  coincidence  with 
that  reflected  by  the  mercury.  The  sextant  reading  is  then 
corrected  for  the  whole  of  the  index  correction.  Half  of  the 
result  will  be,  as  shown  in  Art.  30,  the  apparent  altitude  of 
the  body,  to  which  must  be  applied  the  correction  for  paral- 
lax and  refraction  to  obtain  the  true  altitude.  When  the 
sun  or  the  moon  is  observed,  the  upper  or  the  lower  limb  of 
the  image  reflected  by  the  sextant  is  brought  into  contact 
with  the  opposite  limb  of  the  image  reflected  by  the  mercury, 
and  the  correction  for  semi-diameter  also  is  applied. 

45.  The  Vernier. — The  vernier  is  an  instrument  by  which, 
as  by  the  reading  microscope  previously  explained,  fractions 
of  a  division  of  a  limb  may  be  read.     In  Fig.  17,  let  AB 
be  an  arc  of  a  stationary  graduated  circle,  and  let  CD  be  a 
movable    arm,    carrying    another    graduated    arc    at    its 
extremity.     The  value  of  each  division  of  the  limb  AB  is 
one-sixth  of  a  degree,  or  10'.     The  arc  on  the  arm  CD  is 


THE  VERNIER 


45 


divided  into  ten  equal  parts,  and  the  length  of  the  arc  between 
the  points  0  and  10  is  equal  to  the  length  of  nine  divisions 
of  the  arc  AB.  This  arc,  which  the  limb  CD  carries,  is 
called  a  vernier.  Since  the  ten  divisions  of  the  vernier 
equal  in  length  nine  divi- 
sions of  the  limb,  it  follows 
that  each  division  of  the 
vernier  comprises  9'  of  arc; 
in  other  words,  any  divi- 
sion of  the  vernier  is  less 
by  V  of  arc  than  any  divi- 
sion of  the  limb. 

The  reading  of  any  in- 
strument which  carries  a 
vernier  is  always  deter- 
mined by  the  position  of 
the  zero  point  of  the  vernier.  FIG.  17. 

If,  now,  the  zero  point  of 

the  vernier  exactly  coincides  with  a  division  of  the  limb, 
the  point  of  the  vernier  1  will  fall  V  behind  the  next  division 
of  the  limb,  the  point  2  will  fall  2'  behind  the  next  division 
but  one,  and  so  on;  and  if,  such  being  the  case,  the  vernier 
is  moved  forward  through  an  arc  of  1',  the  point  1  will  come 
into  coincidence  with  a  division  of  the  limb;  if  it  is  moved 
forward  through  an  arc  of  2',  the  point  2  will  come  into 
coincidence  with  a  division  on  the  limb;  and,  in  general,  the 
number  of  minutes  of  arc  by  which  the  zero  point  of  the 
vernier  falls  beyond  the  division  of  the  limb  which  imme- 
diately precedes  it  will  be  equal  to  the  number  of  that  point 
of  the  vernier  which  is  in  coincidence  with  a  division  of  the 
limb.  If,  then,  the  zero  point  falls  between  any  two  divi- 
sions of  the  limb,  as  11°  20'  and  11°  30',  for  example,  and 
the  point  2  of  the  vernier  is  found  to  be  in  coincidence 
with  any  division  of  the  limb,  we  know  that  the  zero  point 
is  2'  beyond  the  division  11°  20',  and  that  the  complete 
reading  for  that  position  of  the  verniers  is  11°  22'. 


46  ASTRONOMICAL  INSTRUMENTS,  ERRORS 

46.  General  Rules  of  Construction.  —  In  the  construction 
of  all  verniers  similar  to  the  one  above  described,  the  same 
rules  of  construction  must  be  followed:  the  length  of  the 
arc  of  the  vernier  must  be  exactly  equal  to  the  length  of  a 
certain  number  (no  matter  what)  of  the  divisions  of  the 
limb,  and  the  arc  must  be  divided  into  equal  parts,  the 
number  of  which  shall  be  greater  by  one  than  the  number 
of  these  divisions  of  the  limb.  Following  these  rules,  and 
putting 

D  =  the  value  of  a  division  of  the  limb, 
d=    "       "        "         "  "      vernier, 

n=the  number  of  equal  parts  into  which  the  vernier  is 
divided, 

we  have 

D-d  =  » 

n 

as  a  general  formula.     The  difference  D—d  is  called  the 
least  count  of  the  vernier. 

If,  in  Fig.  17,  we  take  the  length  of  the  vernier  equal  to 
59  divisions  of  the  limb,  and  divide  it  into  60  equal  parts, 
we  shall  have 


which  is  the  least  count  on  most  of  the  modern  sextants. 

Verniers  are  sometimes  constructed  in  which  the  number 
of  equal  parts  on  the  vernier  is  less  by  one  than  the  number 
of  the  divisions  of  the  limb  taken.  In  this  case  we  have 


and  the  only  difference  between  this  class  of  verniers  and 
the  class  above  described  is  that  the  graduations  of  the 
limb  and  the  vernier  proceed  in  this  class  in  opposite  direc- 
tions. 


OTHER  ASTRONOMICAL  INSTRUMENTS  47 

To  ensure  more  accurate  reading  of  the  latest  high  grade 
sextants,  the  tangent  screw  is  connected  with  a  graduated 
head  similar  to  that  on  the  micrometer. 


OTHER  ASTRONOMICAL  INSTRUMENTS 

47.  The  Zenith  Telescope,  the  Theodolite,  and  the  Univer- 
sal Instrument  are,  in  general  principle,  only  modified  forms 
of  the  portable  altitude  and  azimuth  instrument. 

The  octant  (sometimes  improperly  called  the  quadrant)  is 
identical  in  construction  with  the  sextant,  excepting  only 
that  its  arc  contains  45°. 

The  prismatic  sextant  carries  a  reflecting  prism  in  place 
of  the  ordinary  horizon-glass,  and  the  graduated  arc  com- 
prises a  semi-circumference. 

The  reflecting  circle  is  still  another  modification  of  the 
sextant,  in  which  the  graduated  arc  is  an  entire  circum- 
ference, and  the  index-bar  is  a  diameter  of  the  circle,  revolving 
about  the  center,  and  carrying  a  vernier  at  each  extremity. 
Sometimes  the  circle  has  three  verniers,  at  intervals  of  120° 
of  the  graduated  arc. 

Photography  is  now  extensively  used  in  telescopic 
research.  It  furnishes  a  means  of  studying  not  only  the 
sun,  the  moon,  and  the  planets,  but  also  the  stars,  comets, 
and  nebulae.  It  is  being  used  in  cataloguing  the  stars; 
and  plans  are  now  being  carried  out  which  will  determine 
the  places  of  many  millions  of  stars.  In  stellar  photography 
the  observer's  eye  is  replaced  by  the  camera  at  the  eye-piece 
of  the  telescope;  if  the  telescope  is  moved  by  the  driving 
clock,  as  explained  in  Article  38,  any  portion  of  the  visible 
heavens  can,  under  favorable  circumstances,  be  photo- 
graphed. Stars  within  the  field  of  view  will  be  photographed 
as  points;  planets,  satellites,  etc.,  will  change  their  position 
with  reference  to  the  plate,  if  the  exposure  continue  long 
enough,  and  will  leave  what  are  technically  called  "trails." 
Many  of  the  important  discoveries  of  recent  years  are  the 


48  ASTRONOMICAL  INSTRUMENTS,  ERRORS 

result  of  improved  methods  and  appliances  for  stellar  pho- 
tography. 

ERRORS 

48.  However  carefully  an  instrument  may  be  constructed, 
however  accurately  adjusted,  and  however  expert  the 
observer  may  be,  every  observation  must  still  be  regarded 
as  subject  to  errors.  These  errors  may  be  divided  into  two 
classes,  regular  and  irregular  errors.  By  regular  errors  we 
mean  errors  which  remain  the  same  under  the  same  com- 
bination of  circumstances,  and  which,  therefore,  follow  some 
determinate  law,  which  may  be  made  the  subject  of  inves- 
tigation. Among  the  most  important  of  this  class  of  errors 
are  instrumental  errors:  errors,  that  is  to  say,  due  to  some 
defect  in  the  construction  or  adjustment  of  an  instrument. 
If,  for  instance,  what  we  call  the  vertical  circle  of  the 
meridian  circle  is  not  rigorously  a  circle,  or  is  imperfectly 
graduated;  or  if  the  horizontal  axis  is  not  exactly  horizontal, 
or  does  not  lie  precisely  east  and  west;  any  one  of  these 
imperfections  will  affect  the  accuracy  of  the  observation. 
The  observer,  however,  knowing  what  the  construction  and 
adjustment  of  the  instrument  ought  to  be,  can  calculate 
what  effect  any  given  imperfection  will  produce  upon  his 
observation,  and  can  thus  determine  what  the  observation 
would  have  been  had  the  imperfection  not  existed.  Regular 
errors,  then,  may  be  neutralized  by  determining  and  applying 
the  proper  corrections. 

Irregular  errors,  on  the  contrary,  are  errors  which  are  not 
subject  to  any  known  law.  Such,  for  example,  are  errors 
produced  in  the  amount  of  refraction  by  anomalous  con- 
ditions of  the  atmosphere;  errors  produced  by  the  anoma- 
lous contraction  or  expansion  of  certain  parts  of  the  instru- 
ment, or  by  an  unsteadiness  of  the  telescope  produced  by 
the  wind;  and,  more  particularly,  errors  arising  from  some 
imperfection  in  the  eye  or  the  touch  of  the  observer.  Errors 


ERRORS  49 

such  as  these,  being  governed  by  no  known  law,  can  never 
be  made  the  subject  of  theoretic  investigation;  but  being 
by  their  very  nature  accidental,  the  effects  which  they 
produce  will  sometimes  lie  in  one  direction  and  sometimes 
in  another;  and  hence  the  observer,  by  repeating  his  observa- 
tions, by  changing  the  circumstances  under  which  he  makes 
them,  by  avoiding  unfavorable  conditions,  and  finally  by 
taking  the  mean,  or  the  most  probable  value  of  the  results 
which  his  different  observations  give  him,  can  very  much 
diminish  the  errors  to  which  any  single  observation  would 
be  exposed. 

NOTE. — For  complete  descriptions  of  the  various  astronomical  in- 
struments, the  student  is  referred  to  Chauvenet's  Spherical  and  Prac- 
tical Astronomy;  Loomis's  Practical  Astronomy;  and  Pearson's  Prac- 
tical Astronomy. 


CHAPTER  III 
REFRACTION.    PARALLAX.     DIP  OF  THE  HORIZON 

REFRACTION 

49.  When  a  ray  of  light  passes  obliquely  from  one  medium 
to  another  of  different  density,  it  is  bent,  or  refracted,  from 
its  course.  If  a  line  is  drawn  perpendicular  to  the  surface 
of  the  second  medium  at  the  point  where  the  ray  meets  it, 
the  ray  is  bent  towards  this  perpendicular  if  the  second 
medium  is  the  denser  of  the  two,  and  from  it  if  the  first 
medium  is  the  denser. 

In  Fig.  18,  let  A  A,  BB,  represent  two  media  of  different 
density,  the  density  of  BB  being  the  greater.    Let  CD  be  a 
M    p.    c  ray  of  light  meeting  the  surface  of 

BB  at  D.  At  D  erect  the  line  ND 
perpendicular  to  the  surface  of  BB, 
and  prolong  it  in  the  direction  DM. 
The  ray  CD  is  called  the  incident 


G    M  ray,  and  the  angle  NDC  the  angle 

FIG.  18.  of  incidence.     When  the  ray  enters 

the  medium  BB,  it  will  still  lie  in  the 

same  plane  with  CD  and  ND,  but  will  be  bent  towards  the  line 
DM,  making  with  it  some  angle  GDM,  less  than  the  angle 
NDC.  To  an  observer  whose  eye  is  at  G,  the  ray  will 
appear  to  have  come  in  the  direction  EG,  which  is  therefore 
called  the  apparent  direction  of  the  ray.  DG  is  called  the 
refracted  ray,  and  the  angle  GDM  the  angle  of  refraction. 
The  angle  EDC,  the  difference  between  the  directions  of 
the  incident  and  the  refracted  ray,  is  called  the  refraction. 

50 


REFRACTION  51 

ft  is  shown  in  Optics  that,  whatever  the  angle  of  inci- 
dence may  be,  there  always  exists  a  constant  ratio  between 
the  sine  of  the  angle  of  incidence  and  that  of  the  angle  of 
refraction,  as  long  as  the  same  two  media  are  used  and 
their  densities  are  unchanged.  We  have,  then,  in  the 

figure,  1 

smNDC  = 

smGDM~ 

k  being  a  constant  for  the  two  media  A  A  and  BB. 

If  the  second  medium,  instead  of  being  of  uniform 
density,  is  composed  of  parallel 

strata,  each  one  of  which  is  of         

greater  density  than  the  one  im-  A 

mediately  preceding,  as  is  repre- 

sented  in  Fig.  19,  the  path  of  the        B—        ft   -B 

ray  through  these  several  strata 

will  be  a  broken  line,  Dabc;  and  FIG.  19. 

if  the  thickness  of  each  of  these 

successive  strata  is  supposed  to  be  indefinitely  small,  this 

broken  line  will  become  a  curve. 

In  the  figures  above  used,  the  media  are  represented  as 
separated  by  plane  surfaces;  but  the  same  phenomena  are 
noticed,  and  the  same  laws  hold  good,  if  the  media  are 
separated  by  curved  surfaces. 

50.  Astronomical  Refraction. — It  is  determined  by  exper- 
iment that  the  density  of  the  air  gradually  diminishes  as  we 
ascend  above  the  surface  of  the  earth,  and  it  is  estimated 
that  at  a  distance  of  fifty  miles  above  the  surface  the  density 
of  the  air  is  so  small  that  it  exerts  no  appreciable  refracting 
power,  though  it  has  been  found  by  observation  of  meteors 
that  the  atmosphere  extends  to  a  distance  of  75  miles 
above  the  surface.  We  may,  therefore,  consider  the  au- 
to be  made  up  of  a  series  of  strata  concentric  with 
the  earth's  surface,  the  thickness  of  each  stratum  being 
indefinitely  small,  and  the  density  of  each  stratum  being 
greater  than  that  of  the  stratum  next  above  it.  Now,  in 


52      REFRACTION.    PARALLAX.    DIP  OF  THE  HORIZON 


S' 


FIG.  20. 


Fig.  20,  let  the  arc  BD  represent  a  portion  of  the  earth's 
surface,  and  the  arc  MN  a  portion  of  the  upper  limit  of 
the  atmosphere.  Let  S  be  a  celestial  body,  and  SA  a 
ray  of  light  from  it,  which  enters  the  atmosphere  at  A. 

Let  the  normal  (or  radi- 
us) AC  be  drawn.  As 
the  ray  of  light  passes 
down  through  the  atmos- 
phere, it  is  continually 
passing  from  a  rarer  to 
a  denser  medium,  so  that 
its  path  is  continually 
changed,  and  becomes 
a  curve  AL,  concave 
towards  the  earth,  and 
reaching  the  earth  at 
some  point  L.  Since  the 
direction  of  a  curve  at 
any  point  is  the  direction 

of  the  tangent  to  the  curve  at  that  point,  the  apparent 
direction  of  the  ray  of  light  at  L  will  be  represented  by  the 
tangent  LS',  and  in  that  direction  will  the  body  S  appear 
to  lie,  to  an  observer  at  L.  If  the  radius  CL  be  indefinitely 
prolonged,  the  point  Z,  where  it  reaches  the  celestial  sphere, 
will  be  the  zenith  of  the  observer  at  L,  the  angle  ZLS' 
will  be  the  apparent  zenith  distance  of  the  body  S,  and  the 
angle  which  the  line  drawn  from  the  body  to  the  point  L 
makes  with  the  line  LZ  will  be  the  true  zenith  distance  of  S. 
The  effect,  then,  of  refraction  is  to  decrease  the  apparent 
zenith  distances,  or  increase  the  apparent  altitudes  of  the 
celestial  bodies.  Since  the  incident  ray  SA,  the  curve  AL, 
and  the  tangents  LS'  all  lie  in  the  same  vertical  plane,  the 
azimuth  of  the  celestial  bodies  is  not  affected. 

51.  General  Laws  of  Refraction. — By  an  investigation 
of  the  formulae  of  refraction,  and  by  astronomical  observa- 
tions already  described  (Art.  36),  the  amount  of  refraction 


REFRACTION  53 

at  different  altitudes  has  been  obtained,  and  is  given  in  what 
are  called  "tables  of  refraction."  The  following  general 
laws  of  refraction  will  serve  to  give  the  student  some  idea 
of  its  amount,  and  of  the  conditions  under  which  it  varies: 

(1)  In  the  zenith  there  is  no  refraction. 

(2)  The  refraction  is  at  its  maximum  in  the  horizon, 
being  there  equal  to  about  33'.     At  an  altitude  of  45°  it 
amounts  to  57". 

(3)  For  zenith  distances  which  are  not  very  large,  the 
refraction  is  nearly  proportional  to  the  tangent  of  the  zenith 
distance.     When  the  zenith  distance  is  large,  however,  the 
expression  of  the  law  is  much  more  complicated.     No  table 
of  refraction  can  be  trusted  for  an  altitude  of  less  than  5°, 
because  the  effect  of  irregularities  in  atmospheric  conditions 
near  the  surface  of  the  earth  is  to  cause  great  abnormalities 
in  refraction. 

(4)  The  amount  of  refraction  depends  upon  the  density 
of  the  air,  and  is  nearly  proportional  to  it.     The  table  gives 
the  refraction  for  a  mean  state  of  the  atmosphere,  taken 
with  the  barometer  at  30  inches  and  the  thermometer  at 
50°.     If  the  temperature  remains  constant,  and  the  barom- 
eter stands  above  its  mean  height,  or  if  the  height  of  the 
barometer  is  constant,  and  the  thermometer  stands  below 
its  mean  height,  the  density  of  the  atmosphere  is  increased, 
and  the  refraction  is  greater  than  its  mean  amount.     Supple- 
mentary tables  are  therefore  given,  from  which,  with  the 
observed  heights  of  both  barometer  and  thermometer  as 
arguments,  we  may  take  the  necessary  corrections  to  be 
applied  to  the  mean  refraction. 

(5)  Since   the   effect   of   refraction   is   to   increase   the 
apparent  altitudes  of  the  celestial  bodies,  the  amount  of 
refraction  for  any  apparent  altitude  is  to  be  subtracted  from 
that  apparent  altitude,  or  added  to  the  corresponding  zenith 
distance. 

52.  Effects  of  Refraction.— The  apparent  angular  diam- 
eter of  the  sun  and  of  the  moon  being  about  32',  and  the 


54      REFRACTION.    PARALLAX.    DIP  OF  THE  HORIZON 

refraction  in  the  horizon  being  33',  it  follows  that  when  the 
lower  limb  of  either  body  appears  to  be  resting  on  the 
horizon,  the  body  is  in  reality  below  it.  One  effect,  then, 
of  refraction  is  to  lengthen  the  time  during  which  these 
bodies  are  visible.  Still  another  effect  is  to  distort  the  discs 
of  the  sun  and  the  moon  when  near  the  horizon:  for  since 
the  refraction  varies  rapidly  near  the  horizon,  the  lower 
extremity  of  the  vertical  diameter  of  the  body  will  be  more 
raised  than  the  upper  extremity,  thus  apparently  shortening 
this  diameter,  and  giving  the  body  an  elliptical  shape.  When 
the  body  comes  still  nearer  the  horizon,  its  disc  is  distorted 
into  what  is  neither  a  circle  nor  an  ellipse,  but  a  species  of 
oval,  in  which  the  curvature  of  the  lower  limb  is  less  than 
that  of  the  upper  one.  The  apparent  enlargement  of  these 
bodies  when  near  the  horizon  is  merely  an  optical  delusion, 
which  vanishes  when  their  diameters  are  measured  with  an 
instrument. 

PARALLAX 

53.  The  parallax  of  any  object  is,  in  the  general  sense  of 
the  word,  the  difference  of  the  directions  of  the  straight 
lines  drawn  to  that  object  from  two  different  points:  or  it 
is  the  angle  at  the  object  subtended  by  the  straight  line 
connecting  these  two  points.  In  Astronomy,  we  consider 
two  kinds  of  parallax :  geocentric  parallax,  by  which  is  meant 
the  difference  of  the  directions  ^  of  the  straight  lines  drawn 
to  the  center  of  any  celestial  body  from  the  earth's  center 
and  any  point  on  its  surface,  and  heliocentric  parallax,  or 
the  difference  of  the  directions  of  the  lines  drawn  to  the 
center  of  the  body  from  the  center  of  the  earth  and  the 
center  of  the  sun.  The  former  is  the  angle  at  the  body 
subtended  by  that  radius  of  the  earth  which  passes  through 
the  place  of  observation:  the  latter  the  angle  at  the  body 
subtended  by  the  straight  line  joining  the  center  of  the 
earth  and  that  of  the  sun. 


PARALLAX 


55 


FIG.  21. 


54.  Geocentric  Parallax. — In  Fig.  21,  let  C  be  the  center 
of  the  earth,  and  L  some  point  on  its  surface,  of  which  Z 
is  the  zenith.  Let.  S  be 
some  celestial  body.  The 
geocentric  parallax  of  the 
body  is  the  angle  CSL. 
Let  S'  be  the  same  body 
in  the  horizon.  The  angle 
LS'C  is  the  parallax  of 
the  body  for  that  position, 
and  is  called  its  horizontal 
parallax.  If  we  denote  this 
horizontal  parallax  by  P, 
the  earth's  radius  by  R, 
and  the  distance  of  the  body 
from  the  earth's  center  by  d,  we  have,  by  Trigonometry, 

R 
sin  P  —  -J- 

a 

To  find  the  parallax  for  any  other  position,  as  at  S,  we 
represent  the  angle  LSC  by  p,  and  the  apparent  zenith 
distance  of  the  body,  or  the  angle  ZLS,  by  2,  the  sine  of 
which  is  equal  to  the  sine  of  its  supplement  SLC.  We  have 
from  the  triangle  LSC,  since  the  sides  of  a  plane  triangle 
are  proportional  to  the  sines  of  their  opposite  angles, 

sin  p  _  R 
sin  z     d' 

Combining  this  equation  with  the  preceding,  we  have, 
sin  p  =  sin  P  sin  z. 

Since  P  and  p  are  small  angles,  we  may  consider  them 
proportional  to  their  sines,  and  thus  have,  finally, 

p  =  P  sin  z. 

The  parallax,  then,  is  proportional  to  the  sine  of  the  zenith 
distance,  and  may  be  found  for  any  altitude  when  the 


56       REFRACTION.    PARALLAX.     DIP  OF  THE  HORIZON 

horizontal  parallax  is  known.     It  evidently  decreases  as  the 
altitude  increases,  and  in  the  zenith  becomes  zero. 

55.  Application  of  Parallax. — In  order  that  observations 
made  at  different  points  of  the  earth's  surface  may  be  com- 
pared, they  must  be  reduced  to  some  common  point.     Geo- 
centric parallax  is  applied  to  reduce  any  altitude  observed 
at  any  place  to  what  it  would  have  been  had  it  been  observed 
at  the  earth's  center.     We  see  from  Fig.  21  that  parallax 
acts  in  a  vertical  plane,  and  that  the  zenith  distance  of  the 
body  as  observed  from  the  earth's  center,  or  the  angle  ZCS, 
is  less   than   the   observed   zenith    distance  ZLS,   by  the 
parallax  CSL.     Parallax,  then,  is  always  subtractive  from 
the  observed  zenith  distance,  and  additive  to  the  observed 
altitude. 

The  parallax  above  described,  is  strictly  speaking,  the 
parallax  in  altitude.  There  is  also,  in  general,  a  similar 
parallax  in  right  ascension,  and  in  declination,  formula3  for 
deriving  which  from  the  parallax  in  altitude  are  given  in 
other  works. 

56.  Heliocentric  Parallax. — It  may  sometimes  happen 
that  we  wish  to  reduce  an  observation  from  what  it  was  at 
the  center  of  the  earth  to  what  it  would  have  been  if  it  had 
been  made  at  the  center  of  the  sun.     Fig.   21,  and    the 
formula?  obtained  from  it,  will  apply  equally  well  to  this 
case,  by  making  the  necessary  changes  in  the  description 
of  the  figure  and  in  the  names  of  the  angles. 

Let  S  be  still  a  celestial  body,  but  let  C  be  the  center  of 
the  sun,  and  L  that  of  the  earth.  The  angle  p  will  then 
represent  the  heliocentric  parallax,  and  the  angle  SLC  the 
angular  distance  of  the  body  from  the  sun,  as  measured  from 
the  earth's  center,  or,  as  it  is  called,  the  body's  elongation. 
The  angle  P  will  be  the  greatest  value  of  the  heliocentric 
parallax,  taken  when  the  body's  elongation  from  the  sun  is 
90°,  and  is  called  the  annual  parallax.  We  shall  then  find 
from  the  formulae  of  Art.  54,  that  the  annual  parallax  has 
for  its  sine  the  ratio  of  the  distance  of  the  earth  from  the 


DIP  OF  THE  HORIZON 


57 


FIG.  22. 


sun  to  that  of  the  body  from  the  sun,  and  that  the  parallax 
for  any  other  position  is  the  product  of  the  annual  parallax 
by  the  sine  of  the  body's  elongation. 

DIP  OF  THE  HORIZON 

57.  The  dip  of  the  horizon  is  the  angular  depression  of 
the  visible  horizon  below  the  celestial  horizon.  In  Fig.  22, 
let  HG  be  a  portion  of  the  earth's 
surface,  and  C  the  earth's  center. 
Let  a  radius  of  the  earth,  CA, 
be  prolonged  to  some  point,  D, 
beyond  the  surface,  and  let  an 
observer  be  supposed  to  be  at 
the  point  D.  At  the  point  D 
let  the  line  BD  be  drawn  per- 
pendicular to  the  line  CD,  and 
also  the  line  D  H,  tangent  to  the 
earth's  surface  at  some  point  H. 

If  these  two  lines  be  revolved  about  the  line  CD,  DB  will 
generate  the  plane  of  the  celestial  horizon  (since  we  have 
seen  that  all  planes  passed  perpendicular  to  the  radius 
will,  when  indefinitely  extended,  mark  out  the  same  great 
circle  on  the  celestial  sphere),  and  DH  will  generate  the 
surface  of  a  cone,  which  will  touch  the  earth  in  a  small 
circle.  If  we  disregard  for  the  present  the  effect  of  the 
earth's  atmosphere,  this  small  circle  will  be  the  visible 
horizon  of  the  observer  at  D,  and  fthe  angle  BD  H  will  be 
the  dip.  DA  is  the  linear  height  of  the  observer  at  D. 

Now  let  a  radius,  C  H,  be  drawn  to  the  point  of  tangency 
H.  The  angles  BDH  and  HCD  having  their  sides  mutu- 
ally perpendicular,  are  equal.  Represent  the  angle  HCD 
by  D,  the  earth's  radius  by  R,  and  the  observer's  height, 
AD,  by  h.  In  the  triangle  DHC,  right-angled  at  H,  we 
have, 


JL. 

R+h 


=  cos  D: 


58       REFRACTION.    PARALLAX.    DIP  OF  THE  HORIZON 

since  D  is  small  and  its  cosine  changes  slowly,  we  take, 

cos  D  =  l  —  2  sin2  \  D; 
hence  we  have, 

=  1  —  2  sin2  |D: 


As  the  angle  D  is  small,  we  may  take 


and  as  h  is  very  small  in  comparison  with  R,  we  may  also 
assume  (R-\-h)  to  be  sensibly  equal  to  R.  Making  these 
changes  in  the  above  equation,  and  finding  the  value  of  D, 
we  have 

2        /T 
:sin  l"\2R' 


D 


Substituting  in  this  expression  the  value  of  the  earth's  radius 
in  feet,  we  have,  finally, 


h  being  expressed  in  feet. 

The  dip,  then,  for  a  height  of  one  foot  is  63".8;  and  for 
other  heights  it  is  proportional 
to  the  square  root  of  the  number 
of  feet  in  the  height. 

58.  Effect  of  Atmospheric  Re- 
G  J  fraction.  —  If  the  effect  of  atmos- 
pheric refraction  is  taken  into 
consideration,  the  line  HD  must 
be  a  curved  line,  as  is  represented 
in  Fig.  23.  The  point  H  will  then 
appear  to  lie  in  the  direction  DH't 

to  the  observer  at  D,  and  the  dip  will  be  the  angle  BDHr. 
The  effect,  then,  of  refraction  is  to  decrease  the  dip,  the 

*  The  curved  line  HD  is  tangent  to  the  earth's  surface  at  H. 


FIG.  23.* 


DIP  OF  THE  HORIZON  59 


amount  by  which  it  is  decreased  being  about  ^th  of  the 
whole. 

59.  Application  of  Dip.  —  Tables  have  been  computed  in 
which  may  be  found  the  proper  dip  for  different  heights 
above  the  surface  of  the  earth.  Dip  constitutes  one  of  the 
corrections  which  are  to  be  applied  at  sea  to  the  observed 
altitude  of  a  celestial  body  to  obtain  its  true  altitude:  its 
altitude,  that  is,  above  the  celestial  horizon.  Since  the 
visible  horizon  lies  below  the  celestial  horizon,  this  cor- 
rection is  evidently  subtractive. 


CHAPTER  IV 
THE  EARTH.     ITS    SIZE,  FORM,  AND  ROTATION 

60.  HAVING  seen  what  are  the  construction  and  the 
adjustments  of  the  principal  astronomical  instruments,  to 
what  uses  each  is  adapted,  and  what  corrections  are  to  be 
applied  to  the  observations  which  are  taken,  we  are  now 
ready  to  proceed  to  the  solution  of  some  of  the  many 
questions  of  interest  which  the  study  of  Astronomy  opens 
to  us.  And  first  of  all,  let  us  see  how  astronomical  observa- 
tions will  help  us  to  a  knowledge  of  the  size  and  the  form 
of  the  earth.  The  question  is  one  of  the  first  importance; 
for  upon  the  determination  of  the  size  of  the  earth  depends 
in  a  great  measure,  as  we  shall  see  further  on,  the  deter- 
mination of  the  magnitudes  and  the  distances  of  the  other 
heavenly  bodies.  Having  seen  the  facts  which  seem  to 
point  to  the  conclusion  that  the  earth  is  spherical  in  form, 
we  will  start  with  the  assumption  that  this  conclusion  is 
correct,  and  proceed  to  determine  the  magnitude  of  the 
earth,  regarded  as  a  sphere. 

Now,  we  know  from  Geometry  what  is  the  ratio  between 
the  radius  of  a  sphere  and  the  circumference  of  any  great 
circle  of  that  sphere;  and,  therefore,  if  we  can  obtain  the 
length  of  the  circumference  of  any  great  circle  of  the  earth, 
of  a  meridian,  for  instance,  we  can  at  once  determine  the 
radius  of  the  earth.  And  more  than  this:  if  we  can  measure 
the  length  of  any  known  arc  of  this  meridian,  of  one  degree, 
for  instance,  we  can  compute  the  length  of  the  entire  cir- 
cumference. The  determination  of  the  earth's  radius,  then, 
depends  only  on  our  ability  to  satisfy  these  two  conditions: 

60 


TRIANGULATION 


61 


(1)  We  must  be  able  to  measure  the  linear  distance  on 
the  earth's  surface  between  two  points  on  the  same  meridian. 

(2)  We  must  be  able  to  measure  the  angular  distance 
between  these  same  two  points. 

61.  First  Condition.  A  reference  to  Fig.  24  will  explain 
how  the  first  of  these  two  conditions  may  be  satisfied.  Let 
A  and  G  represent  the  two  points  on  the  same 
meridian  the  distance  between  which  we  wish 
to  measure.  We  have  already  seen  (Art.  35) 
how  an  altitude  and  azimuth  instrument  may 
be  so  adjusted  that  the  sight-line  of  the  tele- 
scope will  lie  in  the  plane  of  the  meridian. 
Let  an  instrument  be  so  adjusted  at  the  point 
A.  Let  some  convenient  station  B  be  taken, 
visible  at  A}  and  let  the  distance  AB  be  care- 
fully measured.  This  distance  is  called  the 
base-line.  Now,  by  means  of  the  telescope, 
adjusted  to  the  plane  of  the  meridian,  let 
some  point  C  be  established  on  the  meridian, 
which  shall  also  be  visible  from  B,  and  let  the 
angles  CAB  and  ABC  be  measured.  We  now 
know  in  the  triangle  A  BC  two  angles  and  the  included  side,  and 
can  compute  the  distances  AC  and  CB.  We  now  take  some1 
suitable  point  D,  measure  the  angles  DCB  and  DBC,  and 
knowing  CB,  we  can  obtain  the  distance  CD.  The  instru- 
ment is  now  taken  to  the  point  C,  and  again  established  in 
the  plane  of  the  meridian,  either  by  the  method  of  Art.  35, 
or  by  sighting  back,  as  it  is  called,  to  A,  or  by  both  methods 
combined.  A  third  point  on  the  meridian,  E,  visible  from 
D,  is  then  selected,  the  angles  ECD  and  EDC  are  measured, 
and  the  distances  EC  and  ED  computed.  This  process  is 
continued  until  the  whole  distance  between  A  and  G  has 
been  obtained. 

This  method  of  measurement  is  called  the  method  of 
triangulation.  The  base-line,  AB,  is  purposely  taken  under 
circumstances  which  favor  its  accurate  measurement,  and 


FIG.  24. 


62       THE  EARTH.    ITS  SIZE,  FORM,  AND  ROTATION 

the  rest  of  the  work  consists  in  the  determination  of  hori- 
zontal angles  which  presents  no  special  difficulty,  and  in 
the  solution  of  triangles  by  computation. 

Two  things  are  to  be  noticed  in  reference  to  these  tri- 
angles. The  first  is  that  in  selecting  the  points  B,  C,  D, 
etc.,  care  must  be  taken  so  to  choose  them  that  the  triangles 
ABC,  BCD,  etc.,  shall  be  nearly  equiangular;  since  triangles 
in  which  there  is  a  great  inequality  of  the  angles  (ill-con- 
ditioned triangles,  as  they  are  called)  will  be  much  more 
likely  to  cause  some  error  in  the  work.  The  second  thing 
to  be  noticed  is  that  these  triangles  are  really  spherical 
triangles,  and  must  therefore  be  solved  by  the  formulae  of 
Spherical  Trigonometry.  If,  for  any  reason,  we  are  forced 
to  take  any  of  the  points  C}  E,  etc.,  off  the  meridian,  the 
corresponding  distances  can  be  reduced  to  the  meridian  by 
appropriate  formulas. 

The  correctness  of  the  result  may  be  tested  by  measuring 
the  distance  GF,  and  comparing  its  measured  length  with 
that  obtained  by  computation.  The  marvelous  accuracy 
of  this  method  of  measurement  is  shown  by  the  fact  that  in 
an  arc  of  the  meridian  measured  by  the  French  at  the  close 
of  the  last  century,  and  which  was  several  hundred  miles  in 
length,  the  discrepancy  between  the  measured  and  the  com- 
puted length  of  the  second  base-line  was  less  than  12  inches. 

62.  Second  Condition. — The  second  condition  requires 
that  the  angle  at  the  center  of  the  earth,  subtended  by  the 
the  arc  of  the  meridian  measured,  shall  be  obtained.  This 
angle  is  evidently  the  difference  of  latitude  of  the  two 
extremities  of  the  arc,  and  therefore  all  that  is  needed  to 
satisfy  this  condition  is  that  the  latitude  of  each  extremity 
shall  be  determined  by  appropriate  observations. 

Instead  of  determining  the  latitude  of  each  place  inde- 
pendently of  the  other,  we  may,  if  we  choose,  obtain  the 
difference  of  latitude  directly,  by  observing  at  each  place 
the  meridian  zenith  distance  of  the  same  celestial  body. 
In  Fig.  25,  let  A  and  G  be  the  two  extremities  of  the  arc, 


SPHEROIDAL  FORM  OF  THE  EARTH        63 

0  the  center  of  the  earth,  and  S  the  celestial  body  on  the 

meridian.     If  Z'  is  the  zenith  of  the 

point  A,  the  meridian  zenith  distance 

of  S  at  A,  reduced  to    the    center 

of  the  earth,  is  the  angle  Z'OS.     In 

the  same  manner  the  true  meridian 

zenith  distance  of  S  at  G  is  the  angle 

ZOS.    The  difference  of  these  two 

zenith  distances,  or  the  angle  ZOZf, 

is  evidently  the  difference  of  latitude 

of  G  and  A. 

If  the  celestial  body  crosses  the 
meridian  between   the   two  zeniths, 

as  at  S',  the  difference  of  latitude  is  numerically  the  sum  of 
the  two  meridian  zenith  distances. 

63.  Results. — By  the  process  above   described,   or  by 
processes  of  a  similar  character,  arcs  of  different  meridians, 
and  in  different  latitudes,  have  been  carefully  measured. 
The  sum  of  the  arcs  thus  measured  is  more  than  60°,  and 
the  length  of  a  degree  of  the  meridian  has  been  found  to  be, 
on  the  average,   69.055  miles.     Multiplying  this  by  360, 
we  obtain  24,860  miles  for  the  circumference  of  a  meridian, 
and  dividing  this  circumference  by  TT  (3.1416)  we  find  the 
length  of  the  earth's  diameter  to  be  7913  miles. 

64.  Spheroidal  Form  of  the  Earth. — One  remarkable  fact 
is  noticed  when  we  compare  the  lengths  of  the  degrees  of 
the  meridian,  measured  in  different  latitudes;   and  that  is, 
that  the  length  of  the  degree  is  not  the  same  at  all  parts  of  the 
meridian,  but  sensibly  increases  as  we  leave  the  equator.     The 
length  of  a  degree  at  the  equator  is  found  to  be  68.7  miles, 
whilst  at  the  poles  it  is  computed  to  be  69.4  miles.     The 
conclusion  drawn  from  this  fact  is  that  the  figure  of  the 
earth  is  not  rigorously  that  of  a  sphere,  since  a  spherical 
form   necessarily   implies    an    absolute    uniformity   in   the 
length  of  a  degree  in  all  parts  of  a  great  circle.     In  order  to 
determine  the  exact  geometrical  figure  of  the  earth,   we 


64      THE  EARTH.    ITS  SIZE,  FORM,  AND  ROTATION 

must  bear  in  mind  that  the  curvature  of  a  line  is  always 
proportional  to  the  change  in  the  direction  of  the  tangents 
drawn  at  successive  points  of  that  line.  Now,  since  the 
altitude  of  the  elevated  pole  at  any  place  is  equal  to  the 
latitude  of  that  place,  it  follows  that  an  advance  towards 
the  pole  of  one  degree  in  latitude  is  accompanied  by  a 
depression  of  one  degree  in  the  plane  of  the  horizon.  If, 
therefore,  in  order  to  effect  a  change  of  one  degree  in  our 
latitude,  we  are  forced  to  advance  a  greater  number  of 
miles  at  the  pole  than  at  the  equator,  we  conclude  that  the 
curvature  of  the  meridian  is  less  at  the  pole  than  at  the  equator. 

Now,  this  same  inequality  in  its 
curvature  is  also  a  peculiarity  of 
the  ellipse:   and  hence  we  infer 
that  the  form  of  the  earth's  me- 
ridians is   not   that  of  a  circle, 
but  that  of  an  ellipse,  as  repre- 
sented in  Fig.  26.     The  axis  of 
the   earth,  Pp,    corresponds    to 
the  minor  axis  of  an  ellipse,  at 
the  extremities  of  which  the  curv- 
ature is    the    least;    and    the   equatorial  diameter    of   the 
earth,  EQ,  corresponds  to  the  major  axis  of  an  ellipse,  at 
the  extremities  of  which  the  curvature  is  the  greatest. 

The  form  of  the  earth,  then,  is  that  of  the  solid  which 
would  be  generated  by  the  revolution  of  an  ellipse  about 
its  minor  axis,  which  solid  is  called  in  Geometry  an  oblate 
spheroid.  A  more  common  but  less  accurate  name  given  to 
the  form  of  the  earth  is  that  of  a  sphere,  flattened  at  the  poles. 
65.  Dimensions  of  the  Earth. — The  following  are  the 
dimensions  of  the  earth  when  its  spheroidal  form  is  taken 
into  consideration.  The  determination  is  that  adopted  by 
the  U.  S.  Naval  Observatory  as  the  basis  of  all  its  calcu- 
lations and  is  called  Hayford's  "Spheroid  of  1909": 

Equatorial  radius 6,378,388.4  meters 3963.34  miles 

Polar  radius. 6,356,909     meters 3949.99  miles 


DENSITY  OF  THE  EARTH  65 

It  must  be  understood  that  this  spheroid  is  an  approxi 
mation    only;     with    the   introduction    of   modern    precise 
methods  minor  irregularities  in  the  earth's  shape  have  been 
brought  to  light  so  that,  in  reality,  the  earth's  surface  does 
not  correspond,  strictly,  to  that  of  any  geometrical  solid. 

The  compression,  or  oblateness  of  an  oblate  spheroid  is 
the  ratio  of  the  difference  between  the  major  and  the  minor 
axis  of  the  generating  ellipse  to  its  major  axis.  The  com- 
pression of  the  earth  is  therefore, 

26.70  1  1  ,, 

297th' 


If  a  and  b  represent  the  semi-major  and  the  semi-minor 
axes  of  the  generating  ellipse,  the  expression  for  the  volume 
of  the  oblate  spheroid  is  f  ira2b.  Substituting  in  this  expres- 
sion the  values  of  a  and  6,  we  find  the  earth's  volume  to 
be  about  260  billions  of  cubic  miles. 

66.  Density  of  the  Earth.  —  There  are  various  methods 
of  determining  the  mean  density  of  the  earth.  The  following 
is  a  brief  summary  of  the  method  of  determining  it  by 
means  of  the  torsion  balance.  This  balance  consists  of  a 
slender  wooden  rod,  supported  in  a  horizontal  position  by 
a  very  fine  wire  at  its  center.  To  the  extremities  of  this 
rod  are  attached  two  small  leaden  balls.  If  left  free  to  move, 
this  horizontal  rod  will  of  course  come  to  rest  when  the  sup- 
porting wire  is  free  from  torsion.  Two  much  larger  leaden 
balls  are  now  brought  near  the  two  suspended  balls,  and  on 
opposite  sides,  so  that  the  attractions  of  both  balls  may 
combine  to  twist  the  wire  in  the  same  direction.  The 
smaller  balls  will  be  sensibly  attracted  by  the  larger  ones, 
and  the  horizontal  rod  will  change  the  direction  in  which  it 
lies.  The  amount  of  this  deflection  is  very  carefully 
measured,  and  from  it  is  computed  the  attraction  which 
the  large  balls  exert  on  the  small  ones.  But  we  know  the 
attraction  which  the  earth  exerts  on  the  small  balls,  it 
being  represented  by  their  weight:  and  we  know  also  the 


66      THE  EARTH.     ITS  SIZE,  FORM,  AND  ROTATION 

dimensions  of  the  earth  and  the  attracting  balls.  Finally, 
we  know  the  density  of  lead :  and  from  these  data  it  is  possi- 
ble to  compute  the  mean  density  of  the  earth. 

A  series  of  over  2000  experiments  of  this  nature  was 
conducted  hi  England,  in  1842,  by  Sir  Francis  Baily.  The 
mean  density  of  the  earth,  obtained  from  these  experi- 
ments, was  5.67:  the  density  of  water  being  the  unit. 
Other  methods  of  determining  the  density  of  the  earth 
have  been  employed,  the  main  principle  in  each  method 
being  the  comparison  of  the  attraction  exerted  by  the  earth 
upon  some  object  with  that  exerted  by  some  other  body, 
whose  density  can  be  ascertained,  upon  the  same  object. 
The  results  of  these  experiments  do.  not  differ  materially 
from  the  results  of  the  experiments  with  the  torsion  balance. 

The  dimensions  and  density  of  the  earth  being  known, 
what  is  commonly  called  its  weight  can  be  computed.  It 
is  found  to  be  about  six  sextillions  of  tons. 

ROTATION  OF  THE  EARTH 

67.  Up  to  this  point  we  have  assumed  the  earth  to  be  at 
rest,   and  the  apparent  diurnal  motions  of  the  heavenly 
bodies  to  be  real  motions.     By  careful  observation  of  the 
sun,  the  moon,  and  the  most  conspicuous  of  the  planets, 
astronomers  have  demonstrated  that  each  of  these  bodies 
rotates  upon  a  fixed  axis.     Analogy,  therefore,  points  to  a 
similar  rotation  of  our  own  planet:   and  besides  this,  there 
are  many  phenomena  which  are  inexplicable  if  the  earth  is 
at  rest,  but  which  are  fully  accounted  for  on  the  supposition 
that  it  rotates  upon  an  axis.    We  will  now  examine  the 
principal  of  these  phenomena. 

68.  The  weight  of  the  same  body  is  not  the  same  in  different 
latitudes.    Careful  experiments  made  in  different  latitudes 
show  that  the  weight  of  the  same  body  is  not  constant  'at 
all  parts  of  the  earth's  surface,  but  increases  with  the  latitude. 
A  body  which  weighs  194  pounds  at  the  equator  will  weigh 


ROTATION  OF  THE  EARTH  67 

195  pounds  if  taken  to  either  pole;  that  is  to  say,  the  weight 
of  any  body  is  increased  by  y^th  of  itself  when  carried 
from  the  equator  to  the  pole.  This  experiment  cannot  be 
made  with  the  ordinary  balances  in  which  bodies  are  weighed: 
since  it  is  obvious  that  the  same  cause,  whatever  it  may  be, 
which  effects  the  weight  of  the  body  will  also  affect  that  of 
the  weights  by  which  it  is  balanced,  and  by  the  same  amount, 
so  that  the  scales  will  still  remain  in  equilibrium.  If, 
however,  we  test  the  weight  of  a  body  (the  force,  that  is 
to  say,  with  which  it  tends  to  the  earth's  center)  by  the  effect 
which  it  has  in  stretching  a  spring,  the  increase  of  weight 
will  be  found  to  be  as  stated  above. 

Part  of  this  increase  of  weight  is  due  to  the  spheroidal 
form  of  the  earth,  since  a  body  when  at  the  pole  is  nearer 
the  center  of  the  earth  than  when  at  the  equator.  The 
amount  of  increase  due  to  this  cause  has  been  calculated  to 
be  about  -g^th;  hence  the  difference  between  T^-th  and 
5~§~oth,  which  is  ^i^th,  still  remains  to  be  accounted  for. 
We  shall  now  see  how  it  is  completely  accounted  for  by  the 
supposition  that  it  is  the  effect  of  the  centrifugal  force 
which  is  induced  by  a  rotation  of  the  earth  upon  its  polar 
axis. 

69.  Centrifugal  Force.  —  The  tendency  which  a  body  has, 
when  revolving  about  any  point  as  a  center,  to  recede  from 
that  center,  is  called  its  centrifugal  force.  The  formula  for 
the  centrifugal  force  may  be  found  in  any  treatise  on 
Mechanics,  and  is  as  follows: 


J  — 

in  which  /  is  the  centrifugal  force,  r  the  radius  of  the  circle 
of  revolution,  m  the  mass  and  t  the  periodic  time,  or  the  time 
in  which  the  revolution  is  performed.  Now,  in  Fig.  27,  let 
the  earth  be  supposed  to  rotate  about  its  polar  axis,  Pp, 
once  in  every  sidereal  day,  which,  as  we  have  already  seen 
(Art.  7),  is  3m.  56s.  less  than  the  mean  solar  day,  and 


68       THE  EARTH.     ITS  SIZE,  FORM,  AND  ROTATION 

therefore   contains   86,164s.     Substituting   this   value   of   t 
in  the  formula  given  above,   and  substituting  for  r   the 
value  of  the  earth's  equatorial  radius 
in  feet,  m  equal  to  unity,  and  com- 
puting the  value  of/,  we  shall  find  that 
the  centrifugal  force  at  the  equator  is 
0.1113  pound.     Now  the  actual  force 
of  gravity  at  the  equator  is  found,  by 
Mechanics,  to  be  32.09  pounds.   If  the 
earth  were  at  rest,  the  force  of  gravity 
at    the    equator   would    evidently  be 
32.09+0.1113    pounds.      Hence    the 
diminution  of  gravity  at  the  equator,  due  to  centrifugal 
force  (in  other  words,  the    loss    of    weight),  is   equal   to 
0.1113  1    , 

32.2013'  Cr  289 

Since  the  periodic  time  (t  in  the  formula)  is  constant 
for  all  places  on  the  earth's  surface,  it  is  evident  from  the 
formula,  that  the  centrifugal  force  at  any  place  L  is  to  the 
centrifugal  force  at  the  equator  as  the  radius  of  revolution 
at  L,  or  LM,  is  to  CQ.  But  we  have  in  the  figure, 

ML      ML  n/rrn  T     J.M    j 

-7977-  =  777- = cos  MLC  =  cos  Latitude. 
C(j      \jLi 

Denoting,  then,  the  centrifugal  force  at  the  equator  by 
C,  and  that  at  L  by  c,  We  have, 

c  =  C  cos  L: 

or  the  centrifugal  force  varies  with  the  cosine  of  the  latitude. 

The  centrifugal  force  at  L  acts  in  the  direction  of  the 
radius  of  revolution  ML.  Let  its  amount  be  represented 
by  LB,  taken  on  LM  prolonged.  This  force  may  be  resolved 
into  two  forces:  LA,  in  the  direction  from  the  center  of 
the  earth,  and  AB,  at  right  angles  to  LA.  The  force  LA, 
being  directly  opposed  to  the  attraction  of  the  earth,  has 
the  effect  of  diminishing  the  weight  of  bodies  at  L,  and 
may  therefore  be  taken  to  represent  the  loss  of  weight  at  L. 


ROTATION  OF  THE  EARTH  69 

Denoting  the  loss  of  weight  by  w,  and  the  centrifugal 
force  at  L  by  c,  as  before,  we  have,  from  the  triangle  ABL, 

w  =  c  cos  L. 
But  we  have  already, 

c  —  C  cos  L: 
.*.     w  =  C  cos2  L. 

Now,  at  the  equator,  as  is  evident  from  the  figure,  the 
whole  effect  of  the  centrifugal  force  is  exerted  to  diminish 
the  weight  of  bodies,  and  C  therefore  also  represents  the  loss 
of  weight  at  the  equator.  We  have,  then,  finally,  that  the 
loss  of  weight  of  a  body  at  any  latitude,  due  to  centrifugal 
force,  is  equal  to  the  product  of  -^^th  of  the  weight  multiplied 
by  the  square  of  the  cosine  of  the  latitude. 

70.  Spheroidal  Form  of  the  Earth  due  to  Centrifugal 
Force. — We  see',  then,  that  the  supposition  that  the  earth 
rotates  upon  its  axis  fully  explains  the  observed  difference 
in  the  weight  of  the  same  body  in  different  latitudes.  But 
this  is  not  all :  for  if  we  assume  that  the  particles  of  matter 
of  which  the  earth  is  composed  were  formerly  in  a  fluid  or 
molten  condition,  and  therefore  free  to  move,  the  spheroidal 
form  of  the  earth  is  itself  a  proof  of  the  earth's  rotation. 
Numerous  experiments  may  be  made  to  show  that,  for  a 
fluid  body  at  rest,  the  form  of  equilibrium  is  that  of  a  sphere : 
and  that,  for  a  fluid  body  which  rotates,  the  form  of  equilib- 
rium is  that  of  a  spheroid,  the  oblateness  of  which  increases 
with  the  velocity  of  rotation.  Knowing  the  volume  and  the 
density  of  the  earth,  and  assuming  the  time  of  rotation  to  be 
twenty-four  sidereal  hours,  it  is  possible  to  calculate  the  form 
of  equilibrium  which  a  fluid  mass  under  these  conditions 
will  assume :  and  this  form  is  found  to  be  that  of  a  spheroid, 
with  an  oblateness  very  nearly  identical  with  the  known 
oblateness  of  the  earth. 


70      THE  EARTH.    ITS  SIZE,  FORM,  AND  ROTATION 

This  tendency  of  a  fluid  mass  to  assume  a  spheroidal  form 
under  rotation  may  also  be  shown  in  Fig.  27.  The  centrif- 
ugal force,  LB,  was  resolved  into  the  two  forces  LA  and  A  B, 
the  former  of  which  forces  has  already  been  discussed.  The 
effect  of  the  latter  force,  A  B,  is  evidently  a  tendency  in  the 
particle  L  to  move  towards  the  equator  EQ;  and  a  similar 
force  acting  upon  all  the  particles  of  matter  on  the  earth's 
surface,  excepting  those  at  the  poles  and  at  the  equator, 
will  cause  them  all  to  move  in  the  direction  of  the  equator, 
and  thus  give  a  spheroidal  form  to  the  mass. 

71.  Trade  Winds. — The  trade  winds  are  permanent  winds 
which  prevail  in  and  sometimes  beyond  the  torrid  zone. 
These  winds  are  northeasterly  in  the  northern  hemisphere 
and  southeasterly  in  the  southern  hemisphere.  The  air 
within  the  torrid  zone  being,  in  general,  subject  to  a  greater 
degree  of  heat  than  the  air  at  other  portions  of  the  earth's 
surface,  rises,  and  its  place  is  filled  by  air  which  comes  in 
from  the  regions  beyond  the  tropics.  If  the  earth  were  at 
rest,  these  currents  of  air  would  manifestly  have  simply  a 
northerly  and  a  southerly  direction.  Now,  we  all  know  that, 
when  we  travel  in  any  direction  on  a  still  day,  or  even  when 
the  wind  is  moving  in  the  same  direction  with  us,  but  with 
less  velocity,  the  wind  seems  to  come  from  the  point  towards 
which  we  are  going.  We  see  from  Fig.  27  that,  if  the  earth 
is  rotating  upon  its  polar  axis,  the  linear  velocity  of  rotation 
decreases  as  the  latitude  increases.  Hence,  the  air  from 
beyond  the  tropics,  having  at  the  start  only  the  linear  velocity 
of  the  place  which  it  leaves,  will,  as  it  moves  towards  the 
equator,  have  continually  a  less  velocity  than  that  of  the 
surface  over  which  it  passes,  and  will  seem  to  come  from  the 
quarter  towards  which  those  places  are  moving.  If,  then, 
the  earth  is  rotating  from  west  to  east,  these  currents  of  air 
will  have  an  apparent  motion  from  the  east,  which  motion, 
when  compounded  with  the  motion  from  the  north  and  the 
south,  before  mentioned,  will  give  us  the  northeasterly 
and  southeasterly  winds  which  we  call  the  Trades. 


ROTATION  OF  THE  EARTH  71 

72.  The  Pendulum  Experiment.* — The  last  and  de- 
cidedly the  most  satisfactory  proof  of  the  earth's  rotation 
which  we  shall  notice  is  that  which  comes  from  the  apparent 
rotation  of  the  plane  of  a  freely  suspended  pendulum,  when 
made  to  vibrate  at  any  point  on  the  earth's  surface  except 
the  equator. 

It  is  an  established  law  in  Mechanics  that  a  pendulum, 
freely  suspended  from  a  fixed  point,  always  vibrates  in  the 
same  plane;  and  also  that  if  we  give  the  point  of  support  a 
slow  movement  of  rotation  about  a  vertical  axis,  the  plane  of 
vibration  will  still  remain  unchanged.  If,  for  instance,  we 
suspend  a  ball  by  a  string,  and,  having  caused  it  to  vibrate, 
twist  the  string,  the  ball  will  rotate  about  the  axis  of  the 
string,  while  the  plane  in  which  it  vibrates  will  not  be  affected. 

Now,  let  us  suppose  that  a  pendulum  is  suspended  at  the 
north  pole,  and  is  made  to  vibrate:  and  let  us  further  suppose 
that  the  earth  rotates  from  west  to  east,  once  in  24  hours. 
The  line  in  which  the  plane  of  vibration  intersects  the  plane 
of  the  horizon  will  move  about  in  the  plane  of  the  horizon, 
in  a  direction  opposite  to  that  in  which  the  earth  is  rotating, 
and  with  an  equal  velocity,  thus  completing  one  revolution 
in  24  hours.  In  Fig.  28,  let  ACBD 
be  the  horizon  of  the  observer  at  the  A/  A 

north  pole,  and  let  the  earth  rotate 
in  the  direction  indicated  by  the 
arrows.  Let  the  pendulum  at  P  be 
set  swinging  in  the  direction  of  some 
diameter,  AB,  of  the  horizon.  At 
the  end  of  an  hour,  the  rotation 
of  the  earth  will  have  carried  this  FlG*  28' 

diameter  to  some  new  position  A  'Br, 
at  the  end  of  the  next  hour  to  some  new  position  A"B", 
etc.,  while  the  pendulum  will  still  swing  in  the  original  direc- 
tion AB.    To  the  observer,  then,  unconscious  of  the  earth's 

*  This  is  called  Foucault's  experiment.  A  full  discussion  of  it  is 
given  in  the  American  Journal  of  Science,  2d  series,  vols.  XII-XIV. 


72      THE  EARTH.    ITS  SIZE,  FORM,  AND  ROTATION 

rotation,  the  plane  of  vibration,  which  really  remains  un- 
changed, will  appear  to  rotate  in  a  direction  opposite  to 
that  in  which  the  earth  is  rotating. 

At  the  south  pole,  under  the  same  suppositions,  a  similar 
phenomenon  will  be  noticed,  except  that  the  plane  of  vibra- 
tion will  apparently  move  in  the  opposite  direction.  Thus, 
if  at  the  north  pole  the  apparent  motion  of  the  plane  is  like 
that  of  the  hands  of  a  clock,  as  we  look  on  its  face,  the  ap- 
parent motion  at  the  south  pole  will  be  the  opposite  to  this. 

Again,  if  a  pendulum  is  made  to  vibrate  in  the  plane  of  a 
meridian  at  the  equator,  there  will  be  no  apparent  change  in 
the  plane  of  vibration,  since  it  will  always  coincide  with  the 
plane  of  the  meridian,  and  hence  the  pendulum  will  continue 
to  swing  north  and  south  during  the  entire  period  of  trie 
earth's  rotation.  The  condition  that  the  pendulum  shall 
here  swing  in  the  plane  of  a  meridian  is  entirely  unnecessary, 
and  is  made  only  for  the  sake  of  illustration;  for  there  will  be 
no  apparent  change  in  the  plane  of  vibration,  whatever  may 
be  the  direction  in  which  the  pendulum  is  made  to  vibrate. 

The  apparent  rotation,  then,  of  the  plane  of  vibration  of 
the  pendulum  is  360°  in  24  hours  at  the  poles,  and  nothing 
at  the  equator.  At  places  lying  between  the  equator  and 
the  poles,  the  apparent  angular  motion  of  the  plane  of  vibra- 
tion will  be  between  these  two  limits;  in  other  words,  less 
than  360°  in  24  hours.  Appropriate  investigations  show  that 
the  apparent  angular  motion  of  the  plane  of  vibration  at  any 
place  in  any  interval  of  time  is  equal  to  the  angular  amount  of 
the  earth's  rotation  in  that  time,  multiplied  by  the  sine  of  the 
latitude  of  the  place.*  Thus,  at  Annapolis,  we  have  for  the 

*  This  formula  may  be  obtained  by  the  principles  of  the  resolution  of 
rotation,  given  in  treatises  on  Mechanics. 
Thus,  in  the  figure,  the  rotation  of  the  point 
L  about  the  axis  of  the  earth,  PO,  may  be 
resolved  into  two  rotations,  one  about  the 
radius  LO,  and  the  other  about  the  radius 
MO,  drawn  perpendicular  to  LO.  If  v  repre- 
sents the  angular  velocity  of  L  about  the  axis  PO  (or  15°  in  one  hour), 


ROTATION  OF  THE  EARTH  73 

angular  motion  in  one  hour, 

15°  sin  38°  59'  -  9°  26': 

so  that  the  plane  of  vibration  will  make  one  apparent  rotation 
at  Annapolis  in  38h.  09m. 

Such  is  the  theory  of  the  pendulum  experiment.  Now, 
numerous  experiments  have  been  made  in  different  latitudes, 
and  in  every  case  an  apparent  rotation  of  the  plane  of  vibra- 
tion from  east  to  west  has  been  observed,  with  a  rate  agreeing 
very  closely  with,  that  demanded  by  the  theory;  and  the 
conclusion  is  irresistible  that  the  earth  rotates  on  its  polar 
axis,  from  west  to  east,  once  in  every  sidereal  day. 

73.  Linear  Velocity  of  Rotation.— Taking  the  equatorial 
circumference  of  the  earth  to  be  24,900  miles,  we  have  a 
linear  velocity  of  over  1000  miles  an  hour,  and  over  17  miles 
a  minute.  This  is  the  velocity  at  the  equator.  The  linear 
velocity  at  other  points  on  the  earth's  surface  is  less  than 
this,  since  the  circumferences  of  the  parallels  of  latitude  are 
less  than  the  circumference  of  the  equator.  Since  the 
circumference  of  any  parallel  is  to  that  of  the  equator  as 
the  radius  of  the  parallel  is  to  the  radius  of  the  equator,  the 
linear  velocity  will  diminish  as  we  leave  the  equator  in  the 
same  ratio  that  the  radii  of  the  successive  parallels  diminish : 
in  the  ratio,  that  is,  of  the  cosine  of  the  latitude,  as  was 
proved  in  Art.  69.  '  For  instance,  the  cosine  of  60°  being 
J,  the  linear  velocity  at  that  latitude  is  only  8J  miles  a  minute. 

and  v'  and  v"  the  angular  velocities  about  the  axes  LO  and  MO,  we 
have,  from  Mechanics, 

v'  =  v  cos  LOP,  and  v" =v  cos  POM. 

Now,  the  rotation  about  the  axis  OM  will  haye  no  effect  in  changing  the 
apparent  position  of  the  plane  of  vibration  of  the  pendulum,  since  it  is 
analogous  to  the  case  at  the  equator  considered  in  the  text;  while  the 
rotation  about  the  axis  LO,  being  analogous  to  the  case  at  the  pole,  will, 
produce  a  similar  effect.  The  apparent  angular  motion,  then,  of  the 
plane  of  vibration  will  be  v  cos  LOP,  or  v  sin  Lat. 


CHAPTER  V 


s'" 


LATITUDE,   LONGITUDE 

LATITUDE 

74.  THE  latitude  of  any  place  on  the  earth's  surface  has 
been  proved,  in  Articles  10  and  11,  to  be  equal  to  either  the 
altitude  of  the  elevated  pole  or  the  declination  of  the  zenith  at 
that  place.     We  shall  now  proceed  to  explain  the  principal 
methods  by  which  either  one  or  the  other  of  these  arcs  may 
be  found. 

75.  First  Method. — Let  Fig.  29  represent  a  projection  of 
the  celestial  sphere  on  the  plane  of  the  celestial  meridian, 

RZHN,  of  some  place.  HR  is  the 
celestial  horizon  at  that  place,  Z  the 
zenith,  P  the  elevated  pole,  and  EQ 
the  equator.  Let  s  represent  some 
circumpolar  star,  whose  declination  is 
known,  at  its  lower  culmination.  Let 
its  meridian  altitude  be  observed, 
and  corrected  for  instrumental  errors 
and  refraction.  (For  all  celestial 
bodies  except  the  sun,  the  moon, 

and  the  planets,  the  corrections  for  parallax  and  semi- 
diameter  will  be  inappreciable.)  To  this  corrected  alti- 
tude add  the  star's  polar  distance,  the  complement  of  the 
star's  known  declination.  The  sum  is  the  altitude  of  the 
elevated  pole,  or  the  latitude. 

If  the  circumpolar  star  is  at  its  upper  culmination,  as  at 
s',  the  polar  distance  is  to  be  subtracted  from  the  corrected 
altitude. 

74 


LATITUDE  75 

If  h'  and  h  denote  the  corrected  altitudes  at  the  upper 
and  the  lower  culmination,  pr  and  p  the  corresponding  polar 
distances,  and  L  the  latitude,  we  have  evidently 

L  =  h'-pf 
L  =  h+p: 
whence  L  =  ±(h'+h)+i(p-p'). 

In  this  formula  the  value  of  the  latitude  does  not  depend  on 
the  absolute  value  of  either  polar  distance,  but  merely  on  the 
change  of  the  polar  distance  between  the  two  transits,  which 
is  usually  so  small  as  to  be  neglected.  This  method,  then, 
is  free  from  any  error  in  the  declination,  and  is  used  at  all 
fixed  observatories. 

76.  Second  Method. — When  the  star  is  at  its  upper 
culmination,  it  will,  in  general,  be  more  convenient  to  find 
the  declination  of  the  zenith  from  the  meridian  zenith  dis- 
tance of  the  star.  Taking  the  star  s',  for  instance,  and  de- 
noting its  meridian  zenith  distance  by  z,  and  its  declination 
by  d,  we  have 

L=ZQ=Qs'-Zs'  =  d-z (a) 

For  the  star  s",  we  have 

L=Zs"+Qs"  =  z+d,   ' (6) 

and  for  the  star  s'" 

L=Zs'"-Qs'"  =  z-d (c) 

From  these  three  formulae  a  general  rule  may  be  deduced, 
applicable  to  the  upper  culmination  of  every  star.  We  notice 
that  in  the  formulae  (a)  and  (6),  where  d  is  positive,  the 
stars  s'  and  s"  are  on  the  same  side  of  the  equator  with  the 
elevated  pole;  that  is  to  say,  their  declinations  have  the  same 
name  as  the  elevated  pole;  while  in  trie  formula  (c)  the  de- 
clination has  the  opposite  name.  We  also  notice  that  in 


76  LATITUDE,  LONGITUDE 

the  formulae  (b)  and  (c),  where  z  is  positive,  the  stars  are  on 
the  opposite  side  of  the  zenith  from  the  elevated  pole;  in 
other  words,  their  bearing  has  the  opposite  name  to  that  of 
the  pole  :  while  the  bearing  of  the  star  sf,  in  the  formula  for 
which  z  is  negative,  has  the  same  name  as  the  elevated  pole. 
The  general  rule,  then,  for  all  these  stars  will  be  the  following: 
If  the  star  bears  south,  mark  the  zenith  distance  north; 
if  it  bears  north,  mark  the  zenith  distance4  south;  mark  the 
declination  north  or  south,  as  the  star  is  north  or  south  of 
the  equator,  and  combine  the  zenith  distance  and  the  declina- 
tion, thus  marked,  according  to  their  names. 

77.  Third  Method.  —  A  very  successful  adaptation  of  the 
preceding  method  is  made  by  using  two  stars  which  culminate 
at  nearly  the  same  time,  but  on  opposite  sides  of  the  zenith, 
as  s'  and  s"  in  Fig.  29.  These  two  stars  are  so  selected  that 
the  difference  of  their  zenith  distances  is  very  small,  and  can 
be  measured  directly  by  means  of  a  micrometer.  By  the 
formulae  of  the  preceding  article  we  have  for  s', 

L  =  d-z, 

and  for  s",  denoting  its  meridian  zenith  distance  and  declina- 
tion by  z'  and  d', 

•  \    L  =  d'+zf, 
whence  we  have, 


The  determination  of  the  latitude  is  thus  made  free  from 
any  error  in  the  graduations  of  the  vertical  circle,  and  de- 
pends only  on  the  known  declinations  of  the  two  stars,  and 
on  the  difference  of  their  zenith  distances.  Errors  in  the 
refraction  are  also  very  nearly  eliminated. 

This  is  the  principle  of  what  is  called  Talcott's  Method,  a 
method  very  commonly  used  by  the  United  States  Coast 
Survey.  The  instrument  employed  is  the  zenith  telescope, 
a  modification  of  the  altitude  and  azimuth  instrument. 
The  two  stars  are  so  selected  that  the  difference  of  their 


LATITUDE  77 

zenith  distances  is  less  than  the  breadth  of  the  field  of  the 
telescope.  The  instrument  is  set  in  the  plane  of  the  meridian 
to  the  mean  of  the  two  zenith  distances,  and  for  the  star 
which  culminates  first.  When  this  star  crosses  the  meridian, 
it  is  bisected  by  the  micrometer  wire,  and  the  micrometer 
is  read.  The  instrument  is  then  turned  180°  in  azimuth, 
and  the  process  is  repeated  with  the  second  star.  The 
difference  of  the  zenith  distances  is  then  obtained  from  the 
difference  of  the  two  micrometer  readings,  and  added  to  the 
half  sum  of  the  two  declinations,  according  to  the  formula. 

78.  Fourth  Method. — When  the  local  time  (either  solar 
or  sidereal)  is  known,  the  latitude  may  be  obtained  from 
altitudes  which  are  not  measured  on 

the  meridian.  Let  Fig.  30  be  a  pro- 
jection of  the  celestial  sphere  on  the 
plane  of  the  horizon.  Z  is  the  zenith 
of  the  place,  P  the  elevated  pole,  PZ 
the  co-latitude,  and  S  a  star,  whose 
altitude  is  measured.  SPZ  is  the 
hour  angle  of  the  star,  which  can  be 
obtained  from  the  local  time  noted  at 
the  instant  the  altitude  is  observed. 

PS  is  the  star's  known  polar  distance.  In  the  triangle 
SPZ,  we  have  the  sides  ZS  and  SP,  and  the  angle  SPZ,  and 
can  therefore  compute  the  value  of  the  co-latitude,  PZ,  by 
the  formulae  of  Spherical  Trigonometry. 

An  analytical  investigation  of  the  formulae  by  which 
this  problem  is  solved  shows  that  errors  in  the  observed  alti- 
tude and  the  time  have  the  less  effect  upon  the  result  the 
nearer  the  body  is  to  the  meridian. 

79.  Methods  of  Finding  the  Latitude  at  Sea. — The  second 
and  the  fourth  of  the  methods  above  described  are  the 
methods  most  commonly  employed  in  finding  the  latitude  at 
sea.     The  sun  is  the  body  which  is  generally  used,  its  altitude 
above  the  sea  horizon  being  measured  with  a  sextant  or  an 
octant.     The  time  of  noon  being  approximately  known,  the 


78  LATITUDE,  LONGITUDE 

observer  begins  to  measure  the  altitude  of  the  lower  limb  of 
the  sun  a  few  minutes  before  noon,  and  continues  to  measure 
it  until  the  sun  ceases  to  rise,  or  "dips,"  as  it  is  called.  The 
greatest  altitude  which  the  sun  attains  is  considered  to  be 
the  meridian  altitude,  although,  rigorously  speaking,  it  is 
not.  The  proper  corrections  for  index-error,  dip,  refraction, 
parallax,  and  semi-diameter  are  next  applied  to  the  sextant 
reading,  and  the  result  is  the  sun's  true  meridian  altitude, 
from  which  the  latitude  is  obtained  by  the  rule  given  in 
Art.  76. 

When  cloudy  weather  prevents  the  determination  of  the 
meridian  altitude  of  either  the  sun  or  any  other  celestial 
body,  an  altitude  obtained  within  an  hour  of  transit,  on 
either  side  of  the  meridian,  may  be  used  to  find  the  latitude 
by  the  fourth  method,  Art.  78,  provided  the  longitude  is 
known  very  closely.  Bowditch's  Navigator  contains  special 
tables  by  which  the  computation,  particularly  when  the  sun 
is  observed,  may  be  greatly  facilitated. 

80.  Reduction  of  the  Latitude. — Owing  to  the  spheroidal 
form  of  the  earth,  the  vertical  line  at  any  point  of  the  surface, 
asZ'O'  in  Fig.  31,  which  corresponds  exactly  with  the  normal 

drawn  at  that  point,  does  not 
coincide  with  the  radius  of  the 
earth,  Z/0,  passing  through  the 
same  point,  excepting  at  the 
equator  and  the  poles.  It  is 
necessary,  then,  in  refined 
observations,  to  distinguish 
between  the  geographical 
zenith,  Z',  the  point  where 
the  vertical  line,  when  pro- 
longed, meets  the  celestial 

sphere,  and  the  geocentric  zenith,  Z,  the  point  in  which 
the  radius  meets  the  sphere.  Since  there  are  two  zeniths, 
there  are  also  two  latitudes:  Z'O'Q,  the  geographical  lati- 
tude, and  ZOQ,  the  geocentric  latitude.  The  geocentric 


LONGITUDE  79 

latitude  is  evidently  smaller  than  the  geographical  by  the 
angle  OLO',  which  is  called  the  reduction  of  the  latitude. 
Formulae  and  tables  for  finding  this  reduction  are  given  in 
Chauvenet's  Astronomy.  Geocentric  latitude  is  considered 
only  when  the  greatest  accuracy  of  result  is  required. 

LONGITUDE 

81.  Let  Fig.  32  represent  a  projection  of  the  celestial 
sphere  on  the  plane  of  the  equinoctial  ABCG.  P  is  the 
projection  of  the  elevated  pole,  and  PG, 
PA,  and  PB  are  projections  of  arcs  of 
great  circles  of  the  sphere  passing  through 
the  pole.  Let  PG  represent  the  projec- 
tion of  the  meridian  of  Greenwich,  PA 
that  of  the  meridian  of  some  other  place 
on  the  earth's  surface,  and  PB  that  of  the 
circle  of  declination  passing  through  some 
celestial  body  S.  Then  will  the  angle  GPA  FlG-  32- 

represent  the  longitude  of  the  meridian 
PA  from  Greenwich,  GPB  will  represent  the  Greenwich 
hour  angle  of  the  body  St  and  APB  will  represent  its  -hour 
angle  from  the  meridian  PA.  The  difference  between  these 
two  hour  angles  is  evidently  equal  to  the  longitude  of  any 
place  on  the  meridian  PA.  The  longitude,  then,  of  any 
place  on  the  earth's  surface  is  equal  to  the  difference  of  the 
hour  angles  of  the  same  celestial  body  at  that  place  and  at 
Greenwich,  at  the  same  absolute  instant  of  time.  When  the 
Greenwich  hour  angle  is  the  greater  of  these  two  hour  angles, 
reckoned  always  to  the  west,  the  longitude  of  the  place  is 
west:  when  it  is  the  smaller,  the  longitude  is  east. 

If  PB  is  the  hour  circle  passing  through  the  sun,  the 
longitude  of  the  place  is  the  difference  of  the  solar  times  at 
the  place  and  at  Greenwich:  if  it  is  the  hour  circle  passing 
through  the  vernal  equinox,  the  longitude  is  the  difference  of 
the  two  sidereal  times.  In  order,  then,  to  determine  the 


80  LATITUDE,  LONGITUDE 

longitude  of  any  place,  we  must  be  able  to  determine  both 
the  local  and  the  Greenwich  time  (either  sidereal  or  solar) 
at  the  same  instant. 

There  are  various  methods  of  obtaining  the  local  time,  one 
of  which  has  already  been  described  (Art.  20).  It  may  be 
noticed  here  that  we  are  always  able,  by  means  of  the  Nau- 
tical Almanac,  to  convert  sidereal  time  into  solar  time,  or 
solar  into  sidereal  (Art.  105).  It  remains,  then,  to  deter- 
mine the  Greenwich  time,  either  sidereal  or  solar,  to  do  which 
several  distinct  methods  may  be  employed. 

82.  Greenwich  Time  by  Chronometers. — If  a  chronom- 
eter is  accurately  regulated  to  Greenwich  time,  that  is  to 
say,  if  the  amount  by  which  it  is  fast  or  slow  at  Greenwich  on 
any  day,  and  its  daily  gain  or  loss,  are  determined  by  obser- 
vation, the  chronometer  can  be  carried  to  any  other  place  the 
longitude  of  which  is  desired,  and  the  Greenwich  time  which 
the  chronometer  gives  can  be  directly  compared  with  the 
time  at  that  place.     This  would  be  a  perfectly  accurate 
method,  if  the  rate  of  the  chronometer  remained  constant 
during  the  transportation ;  but,  in  fact,  the  rate  of  a  chronom- 
eter while  it  is  carried  from  place  to  place  is  very  rarely 
exactly  the  same  that  it  is  while  the  chronometer  is  at  rest. 
By  using  several  chronometers,  however,  and  by  transporting 
them  several  times  in  both  directions  between  the  two  places, 
and  finally  by  taking  a  mean  of  all  the  results,  the  error  may 
be  reduced  to  a  very  minute  amount.     For  instance,  the 
longitude  of  Cambridge,  Mass.,  was  determined  by  means  of 
fifty  chronometers,  which  were  carried  three  times  to  Liver- 
pool and  back,  and  from  them  the  longitude  was  obtained 
with  a  probable  error  of  only  i  of  a  second  of  time. 

83.  Greenwich  Time  by  Celestial  Phenomena. — There 
are  certain  celestial  phenomena  which  are  visible  at  the  same 
absolute  instant  of  time,  at  all  places  where  they  can  be  seen 
at  all.     Such  are  the  beginning  and  the  end  of  a  lunar  eclipse; 
the  eclipses  of  the  satellites  of  the  planet  Jupiter  by  that 
planet;  the  transits  of  these  satellites  across  the  planet's 


LONGITUDE  81 

disc,  and  their  occult ations  by  it.  The  Greenwich  times  at 
which  these  various  phenomena  will  occur  are  computed 
beforehand,  and  are  published  in  the  Nautical  Almanac. 
The  observer,  then,  to  obtain  his  longitude,  has  only  to  note 
the  local  time  at  which  any  one  of  these  phenomena  occurs, 
and  to  compare  that  time  with  the  corresponding  Greenwich 
time  given  in  the  Almanac.  The  difficulty  of  determining 
the  exact  instant  at  which  these  phenomena  occur,  however, 
diminishes  to  some  extent  the  accuracy  of  the  results.  On 
the  other  hand,  the  times  of  solar  eclipses  and  of  occultations 
of  stars  by  the  moon,  although  not  identical  at  different 
places,  can  be  very  accurately  determined:  and  hence  these 
phenomena  are  often  employed  in  obtaining  longitudes 
(Art.  164). 

84.  Formerly  the  method  of  lunar  distances  was  used  in 
determining   the   Greenwich   time;    the   lunar   distance   is 
the  angle  at  the  earth's  center  between  the  moon's  center 
and  any  other  celestial  body.     The  computation  of  lunar 
distances  is  involved  and,  with  present  methods  of  commu- 
nication, is  so  rarely  necessary  that  this  method  of  checking 
chronometer  errors  is  obsolete.     Radio  stations  all  over  the 
world  make  daily  time  signals  and,  as  most  sea-going  vessels 
are  equipped  with  receiving  apparatus,  the  determination  of 
Greenwich  time  has  been  much  simplified. 

85.  Difference  of  Longitude  by  Electric  Telegraph. — 
When  two  stations,  the  difference  of  longitude  of  which  is 
desired,  are  connected  by  telegraph  wire,  the  difference  of 
longitude  may  be  determined  by  means  of  signals  made  at 
either  station,  and  recorded  at  both.     Suppose,  for  instance, 
there  are  two  stations  A  and  By  of  which  A  is  the  more  easterly, 
and  that  each  station  is  provided  with  a  clock  regulated 
to  its  own  local  time.     Let  the  observer  at  A  make  a  signal, 
the  time  of  which  is  recorded  at  each  station.     Let  X  denote 
the  difference  of  longitude  of  the  two  stations,  T  the  local 
time  at  A  at  which  the  signal  is  made,  and   Tf  the  corre- 
sponding time  at  B.     Since  A  is  to  the  east  of  B,  its  time  is 


82  LATITUDE,  LONGITUDE 

greater  at  any  instant  than  that  of  B.  We  have  then,  sup- 
posing the  signal  to  be  recorded  simultaneously  at  the  two 
stations, 

\=T-T'. 

Experience  proves,  however,  that  the  records  of  the  signal 
are  not  exactly  simultaneous,  since  time  is  required  for  the 
electric  current  to  pass  over  the  wire.  In  the  example  above 
given,  then,  if  we  drnote  the  time  required  by  he  electric 
signal  to  pass  from  A  to  B  by  x,  the  time  recorded  at  B  will 
evidently  be,  not  Tf,  but  T'+x;  so  that  the  expression  for 
the  difference  of  longitude  will  be 

\'=T-T'-x. 

Now  let  us  suppose  that  instead  of  the  signal's  being  made 
by  the  observer  at  A,  it  is  made  by  the  observer  at  B,  at  the 
time  Tf.  The  corresponding  time  recorded  at  A  will  not  be 
T,  but  T+x.  In  this  case,  then,  the  expression  for  the 
difference  of  longitude  will  be 


Taking  the  mean  of  the  values  of  X'  and  X",  we  have 


Any  error,  therefore,  which  is  caused  by  the  time  con- 
sumed by  the  electric  current  in  passing  between  two  stations 
is  eliminated  by  determining  the  difference  of  longitude  by 
signals  made  at  both  stations,  and  taking  the  mean  of  the 
results. 

86.  Difference  of  Longitude  by  "Star  Signals."  —  The 
"method  of  star  signals"  is  a  modification  of  the  method 
described  in  the  preceding  paragraph,  which  is  extensively 
used  in  the  United  States  Coast  Survey.  The  principle  on 
which  this  method  rests  is  that,  since  a  fixed  star  makes  one 
apparent  revolution  about  the  earth  in  exactly  twenty-four 


LONGITUDE  83 

sidereal  hours,  the  difference  of  longitude  between  two  merid- 
ians is  equal  to  the  interval  of  sidereal  time  in  which  any  fixed 
star  passes  from  one  of  these  meridians  to  the  other.  The 
clock  by  which  this  interval  of  time  is  measured  may  be 
placed  at  either  station,  or  indeed  at  any  place  which  is  in 
telegraphic  communication  with  both  stations.  Two  chrono- 
graphs, one  at  each  station,  are  connected  with  the  wire  and 
the  clock,  and  upon  them  are  recorded,  by  breaks  in  the 
circuit  as  explained  in  Art.  22,  the  successive  beats  of  the 
clock.  A  transit  instrument  is  adjusted  to  the  meridian  at 
each  station.  As  the  star  crosses  the  several  threads  of  the 
reticule  of  the  transit  instrument  at  the  eastern  station,  the 
observer,  by  means  of  a  break-circuit  key,  records  the  in- 
stants upon  both  chronographs.  The  same  process  is  re- 
peated as  the  star  crosses  the  wires  at  the  western  station. 
Now,  it  is  evident  that  the  elapsed  time  between  the  transits 
at  the  two  meridians  has  been  recorded  upon  each  chrono- 
graph. Each  of  these  values  of  the  elapsed  time  is  to  be 
corrected  for  instrumental  errors,  errors  of  observation,  and 
for  the  gain  or  loss  of  the  clock  in  the  interval ;  and  the  mean 
of  the  two  values,  thus  corrected,  is  taken  as  the  difference 
of  longitude  of  the  two  places. 

By  making  similar  observations  on  several  stars  on  the 
same  night,  by  repeating  the  observations  on  subsequent 
nights,  by  exchanging  observers  and  using  different  clocks, 
and,  finally,  by  taking  a  mean  of  the  results,  a  very  accurate 
determination  of  the  difference  of  longitude  may  be  secured. 

87.  Method  of  Finding  the  Longitude  at  Sea.— The 
method  of  finding  the  longitude  at  sea  which  is  usually  em- 
ployed is  the  method  of  Art.  82.  The  Greenwich  time  ia 
given  by  chronometers  regulated  to  Greenwich  time,  and  the 
local  time  is  obtained  from  the  observed  altitudes  of  celestial 
bodies.  The  sun  is  the  body  the  altitude  of  which  is  most 
commonly  used  for  this  purpose;  but  altitudes  of  the  moon, 
the  most  conspicuous  of  the  planets,  and  the  fixed  stars  may 
also  be  successfully  employed.  At  the  instant  when  the 


84  LATITUDE.,  LONGITUDE 

altitude  of  any  celestial  body  is  observed,  the  time  shown 
by  a  watch  is  noted.  This  watch,  either  shortly  before  or 
after  the  observation,  is  compared  with  the  Greenwich 
chronometer,  and  by  means  of  this  comparison  the  Greenwich 
time  of  the  observation  is  obtained  from  the  time  given  by 
the  watch.  The  necessary  corrections  are  applied  to  the 
sextant  reading  to  obtain  the  body's  true  altitude.  We 
shall  then  have,  in  the  triangle  PZS,  Fig.  33,  the  side  ZS, 
the  zenith  distance  of  the  body,  PS 
its  polar  distance,  obtained  from  the 
Nautical  Almanac,  and  PZ  the  co- 
latitude  of  the  place  of  observation; 
the  latitude  being  determined  by  some 
one  of  the  methods  already  given 
(Art.  79),  and  being  reduced  to  the 
time  of  observation  by  the  run  of 
the  ship  given  by  the  log.  In  the 
triangle  PZS,  then,  having  the  three 

sides  given,  we  can  compute  the  angle  SPZ}  which  is 
the  hour  angle  of  the  body.  From  this  hour  angle  the 
local  time  can  be  readily  found,  from  which,  and  the  Green- 
wich time  already  obtained,  the  longitude  may  be  determined. 
It  can  be  shown,  by  proper  methods  of  investigation,  that 
an  error  in  the  assumed  latitude,  or  in  the  body's  altitude, 
causes  the  less  error  in  the  resulting  hour  angle  the  nearer 
the  body  is  to  the  prime  vertical.  It  is  best,  then,  in  observ- 
ing the  altitude  of  any  celestial  body  for  the  purpose  of  ob- 
taining the  local  time,  to  observe  it  when  the  body  bears 
nearly  east  or  west,  provided  the  altitude  is  not  so  small  as  to 
be  sensibly  affected  by  errors  in  the  refraction.  It  may  also 
be  shown  that  in  selecting  celestial  bodies  for  observations 
of  this  character,  it  is  best  if  the  other  conditions  are  satisfied, 
to  take  those  bodies  which  have  the  smallest  declinations. 
88.  Comparison  of  the  Local  Times  of  Different  Merid- 
ians.— Since  the  local  time,  either  solar  or  sidereal,  is  the 
greater  at  the  more  easterly  of  any  two  meridians,  it  follows 


LONGITUDE  85 

that  a  watch  or  chronometer  which  is  regulated  to  the  time 
of  any  one  meridian  will  appear  to  gain  when  carried  to  the 
west,  and  to  lose  when  carried  to  the  east:  the  amount  of  gain 
or  loss  in  any  case  being  the  difference  of  longitude,  in  time, 
of  the  two  meridians.  A  watch,  for  instance,  which  gives  the 
correct  solar  time  at  Boston  will,  even  if  it  really  is  running 
accurately,  appear  to  gain  nearly  twelve  minutes  when  taken 
to  New  York.  If,  then,  a  watch  which  is  regulated  to  the 
solar  time  of  any  meridian  is  carried  to  the  east,  the  difference 
of  longitude  in  time  between  the  meridian  left  and  that 
arrived  at  must  be  added  to  the  reading  of  the  watch,  to 
obtain  the  time  at  the  second  meridian :  if  it  is  carried  to  the 
west,  the  difference  of  longitude  must  be  subtracted. 


CHAPTER    VI 

THE  SUN.        THE  EARTH'S  ORBIT.     THE  SEASONS.    TWI- 
LIGHT.    THE  ZODIACAL  LIGHT 

89.  The  Ecliptic. — If  a  great  circle  on  any  globe  is  as- 
sumed to  represent  the  celestial  equator,  and  any  point  of 
that  circle  is  taken  to  represent  the  vernal  equinox,  the 
relative  positions  of  all  bodies,  the  right  ascension  and  declina- 
tion of  which  are  known,  can  be  plotted  upon  this  globe,  and 
we  shall  have  a  representation  of  the  celestial  sphere.  The 
poles  of  the  great  circle  will  represent  the  poles  of  the  celestial 
sphere  and  all  great  circles  passing  through  these  poles  will 
represent  circles  of  declination.  We  have  seen,  in  the 
chapter  on  astronomical  instruments,  in  what  manner  the 
right  ascension  and  the  declination  of  any  celestial  body  can 
be  determined  at  any  time  by  observation.  If  we  thus 
determine  the  position  of  the  sun  from  day  to  day,  and  mark 
the  corresponding  points  upon  our  celestial  globe,  we  shall 
find  that  the  sun  appears  to  move  in  a  great  circle  of  the 
sphere  from  west  to  east,  completing  one  revolution  in  this 
circle  in  365d.  6h.  9m.  9.5s.  of  our  ordinary  solar  time. 
This  interval  of  time  is  called  the  sidereal  year.  The  great 
circle  in  which  the  sun  appears  to  move  is  called  the  ecliptic, 
and  the  two  points  in  which  it  intersects  the  celestial  equator 
are  called  the  vernal  and  the  autumnal  equinox. 

Let  Fig.  34  be  a  representation  of  the  celestial  sphere. 
EAQV  is  the  equinoctial,  Pp  is  the  axis  of  the  sphere,  and 
P  the  north  pole.  The  circle  ACVD  represents  the  ecliptic, 
V  the  vernal,  and  A  the  autumnal  equinox.  The  sun  is 
at  the  vernal  equinox  on  the  21st  of  March.  It  thence  moves 

86 


THE  ECLIPTIC 


87 


eastward  and  northward,  and  reaches  the  point  C,  where 
it  has  its  greatest  northern  declination,  on  the  21st  of  June. 
This  point  is  called  the  northern 
summer  solstice.  From  this  point 
it  moves  eastward  and  south- 
ward, passes  the  autumnal  equi- 
nox A  on  the  21st  of  September, 
and  reaches  the  point  Z>,  called 
the  northern  winter  solstice,  on  the 
21st  of  December.  It  thence 
moves  towards  V, which  it  reaches 
on  the  21st  of  March. 

The  obliquity  of  the  ecliptic 
to  the  equinoctial  is  the  angle 

C  VQ,  measured  by  the  arc  CQ.  This  angle  or  arc  is  evidently 
equal  to  the  greatest  declination,  either  north  or  south,  which 
the  sun  attains,  and  is  found  by  observation  to  be  about 
23°  27'. 

90.  Definitions. — The  latitude  of  a  celestial  body  is  its 
angular  distance  from  the  plane  of  the  ecliptic,  measured  on  a 
great  circle  passing  through  its  poles,  and  called  a  circle  of 
latitude.    In  Fig.  34  the  arc  KS  is  the  latitude  of  the  body  S. 
The  longitude  of  a  celestial  body  is  the  arc  of  the  ecliptic  inter- 
cepted between  the  vernal  equinox  and  the  circle  of  latitude 
passing  through  the  body.     Thus  VK  is  the  longitude  of 
the  body  S.    Longitude  is  properly  reckoned  towards  the  east. 

The  hour  circle  which  passes  through  the  solstices,  the 
circle  DHCB,  is  called  the  solstitial  colure.  The  hour  circle 
which  passes  through  the  equinoxes  is  called  the  equinoctial 
colure. 

91.  Signs. — The  ecliptic  is  divided  into  twelve  equal 
parts,  called  signs,  which  begin  at  the  vernal  equinox,  and 
are  named  eastward  in  the  following  order:  Aries,  Taurus, 
Gemini,   Cancer,   Leo,   Virgo,    Libra,    Scorpio,    Sagittarius, 
Capricornus,  Aquarius,  Pisces.     Hence  the  vernal  equinox 
is  called  the  first  point  of  Aries. 


88     THE  SUN.     THE  EARTH'S  ORBIT.     THE  SEASONS 

The  Zodiac  is  a  zone  or  belt  on  the  celestial  sphere,  ex- 
tending about  8°  on  each  side  of  the  ecliptic. 


DISTANCE  OF  THE  SUN  FROM  THE  EARTH 

92.  Relative  Distances  of  the  Earth  and  Venus  from  the 
Sun. — It  is  found  by  observation  that  the  mean  value  of  the 
sun's  angular  semi-diameter  remains  constant  from  year  to 
year,  being  always  16'  2".  Since  any  increase  or  decrease 
in  the  distance  of  the  earth  from  the  sun  will  evidently  be 
accompanied  by  a  corresponding  decrease  or  increase  in  the 
sun's  angular  semi-diameter,  we  conclude  that  the  mean 
distance  of  the  earth  from  the  sun  is  also  constant  from  year 
to  year.  The  distance  of  the  earth  from  the  sun  may  be 
obtained  by  determining  the  sun's  horizontal  parallax  from 
certain  observations  made  upon  the  planet  Venus.  This 
planet  revolves  in  a  nearly  circular  orbit  about  the  sun,  in  a 
plane  only  3°  inclined  to  the  plane  of  the  ecliptic.  Its 
distance  from  the  sun  is  less  than  that  of  the  earth  from  the 
sun,  and  hence  it  sometimes  passes  between  the  earth  and  the 
sun.  and  is  seen  apparently  moving  across  the  sun's  disc. 
This  phenomenon  is  called  a  transit  of  Venus.  As  a  pre- 
liminary to  the  determination  of  the 
earth's  distance  from  the  sun  from 
one  of  these  transits,  it  is  necessary 
to  obtain  the  relative  distances  of 
Venus  and  the  earth  from  the  sun. 
To  do  this,  in  Fig.  35  let  S  be  the  sun, 
E  the  earth,  and  VV  V"V"r  the 
orbit  of  Venus  about  the  sun.  It  is 
evident  that  the  greatest  angular 
distance  (or  elongation)  of  Venus  from 
the  sun,  the  greatest  value,  that  is,  of 
the  angle  VES,  will  occur  when  the  line 
from  the  earth  to  Venus  is  tangent  to 
the  orbit  of  Venus,  as  represented  in  the  figure.  The  orbit 


FIQ.  35. 


DISTANCE  OF  THE  SUN  FROM  THE  EARTH          89 

of  Venus  is  not  really  a  circle,  but  an  ellipse,  and  hence  the 
distance  VS  is  slightly  variable.  So,  also,  is  the  distance 
SE;  hence  the  greatest  elongation  is  also  variable,  being 
found  to  lie  between  the  limits  of  about  45°  and  47°.  As- 
suming its  mean  value  to  be  46°,  we  have  in  the  right-angled 
triangle  VSE, 


Neglecting  the  inclination  of  the  orbit  of  Venus  to  the 
plane  of  the  ecliptic,  we  shall  have,  at  the  time  of  a  transit, 
when  Venus  is  at  V, 

V'E=.28SE. 

Hence  at  the  time  of  a  transit  the  distance  of  Venus  from  the 

sun  is  to  that  of  Venus  from  the  earth  as  about  72  to  28.* 

93.  Transit  of  Venus.—  In  Fig.  36,  let  S  denote  the  center 

of  the  sun,  and  CADN  its  disc:  let  V  be  Venus,  and  E  the 


FIG.  36. 

center  of  the  earth.  Let  HK  be  that  diameter  of  the  earth 
which  is  perpendicular  to  the  plane  of  the  ecliptic,  and  let  an 
observer  be  supposed  to  be  stationed  at  each  extremity.  In 
order  to  simplify  the  explanation,  let  us  neglect  the  rotation 
of  the  earth  during  the  observation,  and  suppose  Venus  to 
move  in  the  plane  of  the  ecliptic.  To  the  observer  at  H, 
Venus  will  appear  to  move  across  the  sun's  disc  in  the  chord 

*  If  we  know  the  periodic  time  of  Venus  and  that  of  the  earth,  the 
ratio  of  the  distances  of  these  two  planets  from  the  sun  can  be  obtained 
by  Kepler's  Third  Law  (Art.  117),  that  "  the  squares  of  the  periodic 
times  of  any  two  planets  are  proportional  to  the  cubes  of  their  mean 
distances  from  the  sun." 


90      THE  SUN.     THE  EARTH'S  ORBIT.     THE  SEASONS 

CD,  and  to  the  observer  at  K,  in  the  chord  A  B.     Regarding 
VHK  and  VFG  as  similar  triangles,  we  have,  by  Geometry, 

FG:  HK  =  GV  :  VH  =  72:  28 


Again,  we  can  obtain  the  angle  which  the  line  FG  subtends 
at  the  earth's  center  in  the  following  manner.  Let  the  ob- 
server at  H  note  the  interval  of  time  in  which  the  planet 
crosses  the  sun's  disc  in  the  chord  CD,  and  the  observer  at 
K  the  interval  in  which  it  moves  through  the  line  AB. 
Since  there  are  tables  which  give  us  the  angular  velocity,  as 
seen  from  the  earth,  both  of  the  sun  and  of  Venus,  we  can 
deduce  the  angles  at  the  earth's  center  subtended  by  the 
chords  FB  and  GD,  and  knowing  also  the  angular  semi- 
diameter  of  the  sun,  in  other  words,  the  angle  at  the  earth's 
center  subtended  by  SB  or  SD,  we  can  compute  the  angles 
at  the  earth's  center  subtended  by  FS  and  GS,  and,  finally, 
the  angle  subtended  by  FG. 

We  have  now  determined  the  angle  subtended  by  the 
line  FG,  at  a  distance  equal  to  that  of  the  earth  from  the  sun, 
and  also  the  ratio  of  FG  to  the  earth's  diameter.  It  is  evi- 
dently easy  to  obtain  from  these  values  the  distance  of  the 
earth  from  the  sun,  and  the  greatest  angle  at  the  sun  sub- 
tended by  the  earth's  radius,  the  latter  being  the  sun's  hori- 
zontal parallax  (Art.  54). 

Although  we  have  assumed  in  this  discussion  that  the 
two  observers  are  stationed  at  the  extremities  of  the  same 
diameter,  it  is  really  only  necessary  that  they  shall  be  at  two 
places  whose  difference  of  latitude  is  large.  The  earth's 
rotation  and  other  things  which  we  have  here  neglected  must 
be  taken  into  consideration  in  the  practical  determination  of 
the  sun's  parallax. 

94.  Distance  of  the  Earth  from  the  Sun.  —  The  last  three 
transits  of  Venus  were  in  1769,  1874,  and  1882;  and,  from 


MAGNITUDE  OF  THE  SUN  91 

observations  in  1769,  the  sun's  horizontal  parallax  was 
determined  to  be  8". 6.  Later  observations  have  given 
slightly  different  results;  other  methods  of  determining  the 
parallax  have  also  been  used.  At  the  Paris  Conference  in 
1896  astronomers  compared  the  various  values  and  agreed 
that  a  value  of  8". 80  was  most  nearly  correct. 

From  Art.  54,  we  have  for  the  distance  in  miles  of  the 
earth  from  the  sun 

d=R  cosec  P  =  3963.3  cosec  8".80  =  92,897,000  miles. 

95.  Size  of  the  Sun. — The  length  of  the  sun's  radius 
can  be  at  once  obtained  as  soon  as  we  know  its  distance 

from  the  earth.  Thus,  in 
Fig.  37  let  S  be  the  center 
of  the  sun,  and  E  that  of 
the  earth.  The  angle  A  ES 
is  the  apparent  semi-diam- 
eter of  the  sun,  which  we 
obtain  by  observation,  its 
mean  value  being,  as  al- 
ready stated,  16'  2".  We  have  then,  in  the  right-angled 
triangle  AES, 

SA  -92,897,000  sin  16'  2"  =  433,000  miles. 

The  sun's  linear  radius,  then,  is  equal  to  nearly  109.5  of  the 
earth's  radii;  and  since  the  volumes  of  spheres  are  propor- 
tional to  the  cubes  of  their  radii,  the  volume  of  the  sun  bears 
to  that  of  the  earth  the  enormous  ratio  of  1,300,000  to  1. 

By  observations  and  calculations  which  will  be  described 
in  the  Chapter  on  Gravitation  (see  Art.  114),  the  mass  of 
the  sun  is  found  to  be  about  327,000  times  that  of  the  earth; 
or  about  670  times  the  sum  of  the  masses  of  .all  the  planets 
of  the  solar  system. 


92     THE  SUN.     THE  EARTH'S  ORBIT.     THE  SEASONS 


THE  EARTH'S  ORBIT 

96.  Revolution  of  the  Earth  about 'the  Sun.— Up  to  this 
point  we  have  spoken  of  the  apparent  annual  motion  of  the 
sun  in  the  ecliptic  from  west  to  east,  as  though  the  earth  were 
really  at  rest,  and  the  sun  revolved  about  it  in  its  orbit.  But 
when  we  take  into  consideration  the  immense  mass  of  the 
sun  compared  with  that  of  the  earth,  we  are  almost  irresist- 
ibly led  to  conclude  that  the 
apparent  annual  revolution  of 
the  sun  is  the  result,  not  of 
the  actual  revolution  of  the 
sun  about  the  earth,  but  of 
that  of  the  earth  about  the 
sun.  Such  a  revolution  of 
the  earth,  from  west  to  east,* 
would  give  to  the  sun  pre- 
cisely that  apparent  motion 
in  the  ecliptic  which  has 
been  observed.  This  may 
be  seen  in  Fig.  38.  Let 
earth's  orbit,  and  the  outer 


FIG.  38. 


S  be  the   sun,  EE'E"   the 

*  Whatever  the  absolute  motion  of  any  celestial  body  moving  in  a 
circle  or  an  ellipse  may  be,  the  appearance  presented  in  that  motion  will 
be  reversed  if  the  spectator  moves  from  one  side  of  the  plane  in  which 
the  motion  is  performed  to  the  other.  Thus,  the  apparent  daily  motion 
to  the  westward  of  any  celestial  body  is  the  same  as  the  motion  of  the 
hands  of  a  clock  as  we  look  upon  its  face,  to  an  observer  who  is  on  the 
north  side  of  the  plane  of  the  diurnal  circle  in  which  the  body  moves,  as 
is  seen  in  any  latitude  north  of  23°  27  in  the  motion  of  the  sun;  while 
the  same  westward  motion  presents  the  opposite  appearance  if  the 
observer  is  to  the  south  of  the  plane  of  motion,  as  may  be  seen  in  these 
latitudes  in  the  case  of  the  Great  Bear.  The  appearance  presented  by 
a  motion  from  west  to  east  is  of  course  the  reverse  of  this;  hence  when 
we  say  that  the  earth  or  any  other  body  moves  about  the  sun  from  west 
to  east,  we  mean  that,  to  an  observer  situated  to  the  north  of  the  plane  of 
motion,  the  body  appears  to  move  in  a  direction  opposite  to  that  in  which 
the  hands  of  a  clock  move, 


THE  EARTH'S  ORBIT  93 

circle  S'S"S'"  the  great  circle  in  which  the  plane  of  the 
ecliptic,  indefinitely  extended,  meets  the  celestial  sphere. 
When  the  earth  is  at  E,  the  sun  will  be  projected  in  S'; 
when  the  earth  is  at  E',  the  sun  will  be  projected  in  S",  etc.; 
that  is  to  say,  while  the  earth  moves  about  the  sun  in  the 
direction  EE'E",  the  sun  will  apparently  move  about  the 
earth  in  the  same  direction,  S'S"S'". 

Almost  every  celestial  body  in  which  any  motion 
at  all  can  be  detected  is  found  to  be  revolving  about  some 
other  body,  larger  than  itself.  The  moon  revolves  about 
the  earth;  the  satellites  of  the  planets  revolve  about  the 
planets;  and  the  planets  themselves,  some  of  which  are  much 
larger  than  the  earth,  and  at  a  much  greater  distance  from  the 
sun,  revolve  about  the  sun.  There  are  several  observational 
proofs  of  the  revolution  of  the  earth  around  the  sun;  such 
as  the  aberration  of  light,  the  annual  parallaxes  of  stars 
and  spectroscopic  observation  of  stars.  Henceforward,  then, 
we  shall  include  the  earth  in  the  list  of  planets,  and  con- 
sider the  sidereal  year  to  be  the  interval  of  time  in  which 
the  earth  makes  one  complete  revolution  about  the  sun. 

97.  Linear  Velocity  of  the  Earth  in  its  Orbit. — The  num- 
ber of  miles  in  the  circumference  of  the  earth's  orbit,  con- 
sidered as  a  circle,  is  obtained  by  multiplying  the  radius  of  the 
orbit  by  2w.    If  we  then  divide  this  product  by  the  number 
of  seconds  in  a  year,  we  shall  have,  in  the  quotient,  the  num- 
ber of  miles  through  which  the  earth  moves  about  the  sun 
in  a  second  of  time.    It  will  be  found  to  be  about  18.4  miles. 

98.  Elliptical  Form  of  the  Earth's  Orbit.— Although,  as 
has  already  been  stated,  the  mean  value  of  the  sun's  angular 
semi-diameter  remains  constant  from  year  to  year,  careful 
measurements  of  the  semi-diameter  show  that  it  varies  in 
magnitude  during  the  year,  being  greatest  about  the  first  of 
January,  and  least  about  the  first  of  July.     The  evident 
conclusion  from  this  fact  is  that  the  distance  between  the 
earth  and  the  sun  also  varies  during  the  year,  being  greatest 
when  the  sun's  semi-diameter  is  the  least,  and  least  when  it  is 


94      THE  SUN.    THE  EARTH'S  ORBIT.    THE  SEASONS 

the  greatest.     The  truth  of  this  conclusion  may  be  seen  in 
Fig.  37,  in  which  we  have 


As  AS  of  course  remains  constant,  ES  will  vary  inversely  as 
sin  AES,  or  since  the  sines  of  small  angles  are  proportional 
to  the  angles  themselves,  inversely  as  the  angle  AES  itself. 
The  greatest  angular  semi-diameter  of  the  sun  is  16'  17".8, 
the  least  is  15'  45". 7:  hence  the  ratio  of  tiie  greatest  to  the 
least  distance  is  that  of  16'  17". 8  to  15'  45".7,  or  of  1.034  to  1. 
Let  us  now  assume  any  line,  SA  in  Fig.  39,  for  instance,  as 
our  unit  of  measure,  and  prolong  it  until  SH  is  to  SA  as 

1.034  is  to  1.  Then  if  S  de- 
notes the  sun,  SA  and  SH  will 
represent  the  relative  distances 
of  the  earth  from  the  sun  on 
about  the  first  of  January  and 
the  first  of  July.  On  certain 
days  throughout  the  year,  let 
the  advance  of  the  sun  in  longi- 
tude since  the  time  when  the 
earth  was  at  A  be  determined, 

and  let  the  angular  semi-diameter  of  the  sun  on  each  of  these 
days  be  measured.  Lay  off  the  angles  ASB,  ASC,  etc., 
equal  to  these  advances  in  longitude.  Since,  as  may  readily 
be  seen  in  Fig.  38,  the  apparent  advance  of  the  sun  in  longi- 
tude is  caused  by  the  advance  of  the  earth  in  its  orbit,  and  is 
equal  to  it,  the  angles  ASB,  ASC,  etc.,  will  represent  the 
angular  distances  of  the  earth  from  the  point  A  on  the  days 
when  the  different  observations  were  made.  Let  us  next 
take  the  lines  SB,  SC,  etc.,  of  such  lengths  that  each  line  may 
be  to  SA  in  the  inverse  ratio  of  the  corresponding  semi- 
diameters.  If,  then,  we  draw  a  line  through  the  points 
A,  B,  C,  etc.,  we  shall  have  a  representation  of  the  orbit  of 
the  earth  about  the  sun.  The  curve  is  found  to  be  an  ellipse, 


THE  SEASONS  95 

the  sun  being  at  one  of  the  foci.  The  point  A,  where  the 
earth  is  nearest  to  the  sun,  is  called  the  perihelion,  the  point 
H,  the  aphelion,  and  the  angular  distance  of  the  earth  from 
its  perihelion  is  called  its  anomaly. 

The  eccentricity  of  the  ellipse,  or  if  0  is  the  center  of  the 
ellipse,  the  ratio  of  OS  to  OA,  is  evidently  equal  to  about 

or  — rth.    A   more  accurate  value  of  it  is  .0167431. 


1.017        60 

This  eccentricity  is  at  present  subject  to  a  diminution  of 
.000041  a  century;  but  Leverrier,  a  French  astronomer,  has 
proved  that  after  the  eccentricity  has  diminished  to  a  certain 
point  it  will  begin  to  increase  again. 

THE  SEASONS 

99.  The  change  of  seasons  on  the  earth  is  caused  by  the 
inequality  of  the  days  and  nights,  and  this  inequality  is  a 
result  of  the  inclination  of  the  plane  of  the  equinoctial  to  that 
of  the  ecliptic.  The  relative  positions  of  the  sun  and  the 
earth  at  different  parts  of  the  year  are  represented  in  Fig.  40. 
S  represents  the  sun,  and  A  BCD  the  orbit  of  the  earth.  Pp 
is  the  axis  of  rotation  of  the  earth,  and  EQ  the  equator.  The 
plane  of  this  equator  is  supposed  to  intersect  the  plane  of  the 
ecliptic  in  the  line  of  equinoxes  AC  and  to  make  an  angle  of 
23°  27'  with  it.  Since,  as  we  have  already  seen,  the  sun 
appears  to  be  on  this  line  on  the  21st  of  March  and  the  21st 
of  September,  the  earth  itself  must  also  be  on  this  line  at  the 
same  time.  Suppose,  then,  the  earth  to  be  at  A  on  the  21st 
of  March.  The  sun  will  evidently  lie  in  the  direction  AS,  and 
will  be  projected  on  the  celestial  sphere  at  the  vernal  equinox. 
Now,  since  a  line  which  is  perpendicular  to  a  plane  is  per- 
pendicular to  every  line  in  that  plane  which  is  drawn  to  meet 
it,  the  axis  Pp  at  A,  being  perpendicular  to  the  equator,  is 
also  perpendicular  to  the  line  AS,  which  is  common  to  both 
the  plane  of  the  equator  and  the  plane  of  the  ecliptic.  Half 
of  each  parallel  of  latitude  on  the  earth  will  therefore  lie  in 


96     THE  SUN.    THE  EARTH'S  ORBIT.    THE  SEASONS 

light  and  half  in  darkness;  and  hence,  as  the  earth  rotates  on 
the  axis  Pp,  every  point  on  its  surface  will  describe  half  of 
its  diurnal  course  in  light  and  half  in  darkness :  in  other  words, 
day  and  night  will  be  equal  over  the  whole  earth.  Since 
the  direction  of  the  axis  of  rotation  remains  unchanged,  the 
same  condition  of  things  will  occur  when  the  earth  is  at  C,  on 


FIG.  40. 


the  21st  of  September.  Let  the  earth  be  at  B  on  the  21st 
of  June.  Here  we  see  that,  as  the  earth  rotates  on  its  axis 
Pp,  every  point  on  its  surface  within  the  circle  ab  will  lie 
continually  in  the  light,  and  will  hence  have  continual  day, 
while  within  the  corresponding  circle  a'b'  the  night  will  be 
continual.  We  see  also  that  at  the  equator  the  days  and 
nights  will  be  equal,  and  that  every  point  between  the 
equator  and  the  circle  ab  will  describe  more  of  its  diurnal 


THE  SEASONS  97 

course  in  light  than  in  darkness,  and  will  thus  have  its  days 
longer  than  its  nights;  while  between  the  equator  and  the 
circle  a'bf  the  nights  will  be  longer  than  the  days.  Similar 
phenomena  will  occur  when  the  earth  is  at  D,  on  the  21st  of 
December,  except  only  that  it  will  then  be  the  southern 
hemisphere  in  which  the  days  are  longer  than  the  nights  and 
the  southern  pole  at  which  the  sun  is  continually  visible. 

Such,  then,  is  the  inequality  of  the  days  and  nights  caused 
by  the  inclination  of  the  plane  of  the  equinoctial  to  that  of  the 
ecliptic.  As  the  sun  apparently  moves  from  either  equinox, 
the  inequality  of  day  and  night  continually  increases,  reaches 
its  maximum  when  the  sun  arrives  at  either  solstice,  and 
then  continually  decreases  as  the  sun  moves  on  to  the 
equinox:  the  day  being  longer  than  the  night  in  that  hemi- 
sphere which  is  on  the  same  side  of  the  equator  with  the  sun. 
Now,  any  point -on  the  earth's  surface  receives  heat  during  the 
day  and  radiates  it  during  the  night:  and  hence,  when  the 
days  are  longer  than  the  nights,  the  amount  of  heat  received 
is  greater  than  the  amount  radiated,  and  the  temperature 
increases;  while,  on  the  contrary,  when  the  days  are  shorter 
than  the  nights,  the  temperature  decreases:  and  thus  is 
brought  about  the  change  of  seasons  on  the  earth. 

Another  fact,  depending  on  the  same  cause,  and  tending  to 
the  same  result,  must  also  be  taken  into  consideration;  and 
that  is  that  the  temperature  at  any  place  depends  on  the 
obliquity  of  the  sun's  rays:  on  the  altitude,  in  other  words, 
which  the  sun  attains  at  noon.  Now  we  have,  from  Art.  76, 

z  =  L-d: 

from  which  we  see  that,  the  latitude  remaining  constant,  the 
sun  attains  the  greater  altitude,  the  greater  its  declination 
when  it  has  the  same  name  as  the  latitude,  and  the  less  its 
declination  when  it  has  the  opposite  name:  so  that  the 
nearest  approach  to  verticality  in  the  sun's  rays  will  occur  at 
the  same  time  that  the  day  is  the  longest.  An  exception, 
however,  must  be  noticed  to  this  general  rule,  in  the  case  of 


98     THE  SUN.     THE  EARTH'S  ORBIT.     THE  SEASONS 

places  within  the  tropics:  since  at  these  places,  as  may  be 
seen  from  the  formula,  the  sun  passes  through  the  zenith 
when  its  declination  is 'equal  to  the  latitude,  and  has  the 
same  name. 

100.  Effect  of  the  Ellipticity  of  the  Earth's  Orbit  on  the 
Change  of  Seasons. — The  elliptic  form  of  the  earth's  orbit 
has,  perhaps,  little  to  do  with  the  change  of  seasons.  For 
although  the  earth  is  nearer  to  the  sun  on  the  1st  of  January 
than  on  the  1st  of  July,  yet  its  angular  velocity  at  that  time  is 
found  by  observation  to  be  greater,  and  to  vary  throughout 
the  whole  orbit  inversely  as  the  square  of  the  distance.  Now 
it  may  readily  be  shown  that  the  amount  of  heat  received 
by  the  earth  at  different  parts  of  its  orbit  also  varies,  other 
things  being  equal,  inversely  as  the  square  of  the  distance:  so 
that  equal  amounts  of  heat  are  received  by  the  earth  in 
passing  through  equal  angles  of  its  orbit,  in  whatever  part  of 
its  orbit  those  angles  may  be  situated.  Still,  although  the 
change  in  distance  may  not  materially  affect  the  annual 
change  of  seasons,  it  does  affect  the  relative  intensities  of 
the  northern  and  the  southern  summer.  The  southern 
summer  takes  place  when  the  earth's  distance  is  only  about 
firths  of  what  it  is  at  the  time  of  the  northern  summer:  hence, 
the  intensity  at  the  former  period  will  be  to  that  at  the  latter 
in  the  ratio  of  about  (ft)2  to  1,  or  about  yf  to  1:  in  other 
words,  the  intensity  of  the  heat  of  the  southern  summer  will 
be  TTTth  greater  than  that  of  the  heat  of  the  northern  summer. 
Geographical  differences  between  the  two  hemispheres 
modify  this  ratio. 

TWILIGHT 

101.  If  the  earth's  atmosphere  did  not  contain  particles 
of  dust  and  vapor,  which  serve  to  reflect  the  rays  of  light, 
the  transition  from  day  to  night  would  be  instantaneous,  and 
the  intermediate  phenomenon  of  twilight  would  have  no 
existence.  This  phenomenon  is  explained  in  Fig.  41,  in  which 


TWILIGHT  99 

A  BC  represents  a  portion  of  the  earth's  surface,  and  EDF  a 
portion  of  the  atmosphere.  Let  the  sun  be  supposed  to  lie 
in  the  direction  AS,  and  to  be  in  the  horizon  of  the  place  A. 
All  of  the  atmosphere  which  lies  above  the  horizontal  plane 
SD  will  then  receive  the  direct  rays  of  the  sun,  and  A  will 
receive  twilight  from  the  whole  sky.  The  point  B  will,  on 
the  contrary,  be  illuminated  only  by  the  smaller  portion  of 
the  atmosphere  included  within  the  planes  EB  and  AD  and 
the  curved  surface  ED',  and  at  the  point  C  the  twilight  will 
have  wholly  ceased.  Strictly  speaking,  the  lines  AS,  BE, 


FIG.  41. 

etc.,  should  be  slightly  curved,  owing  to  the  effects  of  refrac- 
tion, but  the  omission  involves  no  change  in  the  explanation. 
It  is  computed  that  twilight  ceases  when  the  sun  is  about 
18°  below  the  horizon,  measured  on  a  vertical  circle.  The 
more  nearly  perpendicular  to  the  horizon  is  the  diurnal  circle 
in  which  the  sun  appears  to  move,  the  more  rapid  will  be  the 
sun's  descent  below  the  horizon;  hence,  the  length  of  twilight 
diminishes  as  we  approach  the  equator  and  increases  as  we 
recede  from  it.  Furthermore,  we  see  in  Fig.  2  that  the 
greater  the  declination  of  the  sun,  the  smaller  is  the  apparent 
diurnal  circle  in  which  it  moves,  and  the  greater  will  be  the 
length  of  time  required  for  the  sun  to  reach  the  depression 
of  18°  below  the  horizon.  The  shortest  twilight,  therefore, 
occurs  at  places  on  the  equator,  when  the  sun  is  on  the 
equinoctial,  and  its  length  is  then  Ih.  12m.  Near  the  poles 
the  length  of  twilight  is  at  times  very  great.  Dr.  Hayes,  in 
his  last  expedition  towards  the  North  Pole,  wintered  at 


100   THE  SUN.     THE  EARTH'S  ORBIT.     THE  SEASONS 

latitude  78°  18'  N.,  so  far  above  the  circle  ab,  Fig.  40,  that 
the  sun  was  continually  below  the  horizon  from  the  middle 
of  October  to  the  middle  of  February,  but  at  the  beginning 
and  the  end  of  this  interval  twilight  lasted  for  about  nine 
hours.  At  the  poles  twilight  lasts  nearly  a  month  and  a 
half. 

GENERAL  DESCRIPTION  OF  THE  SUN 

102.  The  surface  of  the  sun  and  the  phenomena  of  sun- 
light are  the  objects  of  assiduous  observation,  and  various 
theories  are  held  by  eminent  astronomers  as  to  the  composi- 
tion of  the  sun  and  to  explain  solar  phenomena.  The 
luminous  surface  or  disc  which  we  see  when  we  observe  the 
sun  is  called  the  photosphere;  "it  is  probably  a  sheet  of  lumi- 
nous clouds  formed  by  condensation  into  little  drops  and 
crystals  of  certain  substances  which,  within  the  central  mass 
of  the  sun,  exist  in  a  gaseous  form,  but  are  cooled  at  its  surface 
below  the  temperature  necessary  for  condensation"  (Young). 
Next  above  the  photosphere  is  a  "reversing  layer"  of  unknown 
thickness  which  contains  the  vapors  of  many  of  the  terrestrial 
chemical  elements.  Outside  of  the  reversing  layer  and  per- 
haps indistinguishable  from  it  is  the  chromosphere,  composed 
of  incandescent  gases  of  which  hydrogen  is  the  most  conspic- 
uous. Still  higher  is  the  corona,  visible  only  during  total 
eclipses  of  the  sun;  this  is  made  up  of  gases  less  dense  than 
those  in  the  other  parts  of  the  sun's  envelope. 

When  the  sun  is  observed  with  a  telescope,  spots  are 
noticed  upon  its  surface.  These  spots  appear  to  cross  the 
disc  from  east  to  west,  and  with  different  rates,  the  rate 
of  motion  of  spots  at  the  sun's  equator  being  the  greatest. 
Professor  Young  of  Princeton,  says  that  "the  probability  is 
that  the  sun,  not  being  solid,  has  really  no  one  period  of 
rotation,  but  that  different  portions  of  its  surface  and  of 
its  internal  mass  move  at  different  rates,  and  to  some  "extent 
independently  of  each  other."  The  period  of  this  rotation  is 


GENERAL  DESCRIPTION  PF^  TH5J  3? £$  V  i  /    101 

about  25  days,  and  the  plane  of  rotation  is  inclined  7°  to  the 
ecliptic.  Much  uncertainty  exists  as  to  the  causes  of  the 
sun  spots.  The  latest  observations  show  that  the  spots  are 
cyclonic  disturbances  in  the  photosphere  and  that  they  are 
rilled  with  vapors  and  gases  that  are  so  much  cooler  than 
the  surrounding  portions  that  they  absorb  a  large  proportion 
of  light.  All  sun  spots  exhibit  a  strong  magnetic  field  in  the 
center;  the  polarity  seems  to  bear  an  analogy  to  high  and 
low  pressure  areas  in  terrestrial  cyclonic  storms.  These  spots 
are  for  the  most  part  confined  to  a  zone  extending  about 
35°  on  each  side  of  the  sun's  equator.  They  differ  widely 
in  duration,  sometimes  lasting  for  several  months  and  some- 
times disappearing  in  the  course  of  a  few  hours.  They  are 
sometimes  of  an  immense  size.  One  was  seen  in  1843,  with  a 
diameter  of  nearly  75,000  miles:  it  remained  in  sight  for  a 
week,  and  was  visible  to  the  naked  eye.  In  1858  a  much 
larger  one  was  seen,  its  diameter  being  over  140,000  miles. 
As  a  general  thing,  each  dark  spot,  or  umbra,  as  it  is  called, 
has  within  it  a  still  darker  point,  called  the  nucleus,  and  is 
surrounded  by  a  fringe  of  a  lighter  shade,  called  the  penumbra. 
Sometimes  several  spots  are  inclosed  by  the  same  penumbra 
and  occasionally  spots  are  seen  without  any  penumbra  at  all. 

One  very  curious  and  interesting  discovery  in  relation  to 
these  spots  is  that  of  a  periodicity  in  their  number.  This 
discovery  was  made  by  Schwabe,  of  Dessau,  whose  researches 
and  observations  on  this  subject  covered  a  period  of  about 
forty-three  years.  The  number  of  groups  of  spots  which  he 
observed  in  a  year  varied  from  33  to  333,  the  average  being 
not  far  from  150.*  He  found  the  period  from  one  maximum 
to  another  to  be  about  ten  years.  Professor  Wolf,  of  Zurich, 
after  tabulating  all  the  observations  of  spots  since  1611, 
decided  that  the  period  varied  from  eight  to  sixteen  years, 
its  mean  value  being  about  eleven  years. 

It  is  a  curious  fact  that  magnetic  storms  and  the  phenom- 
enon called  Aurora  or  Northern  Lights  have  a  similar  period, 
*  A  table  of  Schwabe's  observations  is  given  in  the  Appendix. 


102  THE  SUN.. '  THE  EARTH'S  ORBIT.     THE  SEASONS 

and  are  most  frequent  and  most  striking  when  the  number  of 
the  solar  spots  is  the  greatest. 

Still  other  phenomena  which  are  seen  upon  the  sun's 
disc  are  the  }acul(p,  which  are  streaks  of  light  seen  for  the 
most  part  in  the  region  of  the  spots,  and  which  are  undoubt- 
edly elevations  or  ridges  in  the  photosphere ;  and  the  granu- 
lations, which  are  specks  of  light  scattered  over  the  sun's 
disc,  giving  it  an  appearance  not  unlike  that  of  the  skin  of 
an  orange,  though  relatively  much  less  rough.  The  cause 
of  these  granulations  is  unknown. 

At  the  time  of  a  total  eclipse  of  the  sun  by  the  moon,  the 
disc  of  the  sun  is  observed  to  be  surrounded  by  a  ring  or  halo 
of  light,  which  is  called  the  corona.  The  breadth  of  this 
corona  is  more  than  equal  to  the  diameter  of  the  sun.  Rose- 
colored  protuberances,  sometimes  called  red  flames,  are  also 
seen,  which  are  usually  of  a  conical  shape,  and  are  some- 
times of  great  height.  In  the  total  eclipse  of  August  17th, 
1868,  one  was  observed  with  an  apparent  altitude  of  3', 
corresponding  to  a  height  of  about  80,000  miles.  These 
protuberances  were  formerly  supposed  to  be  similar  in  char- 
acter to  our  terrestrial  clouds;  but  Dr.  Jannsen,  the  chief  of 
the  French  expedition  sent  out  to  the  East  to  observe  the 
total  eclipse  of  August,  1868,  examined  their  light  with  the 
spectroscope,  and  found  them  to  be  masses  of  incandescent 
gas,  of  which  the  greater  part  was  hydrogen.  Dr.  Jannsen 
also  made  the  interesting  discovery  that  these  protuberances 
can  be  examined  at  any  time,  without  waiting  for  the  rare 
opportunity  afforded  by  a  total  eclipse.  He  observed  them 
for  several  successive  days,  and  found  that  great  changes  took 
place  in  their  form  and  size.  Mr.  Lockyer,  of  England,  who 
has  since  examined  them,  pronounces  them  to  be  merely 
local  accumulations  of  a  gaseous  envelope  completely  sur- 
rounding the  sun:  the  spectrum  peculiar  to  these  protuber- 
ances appearing  at  all  parts  of  the  disc. 

It  has  already  been  stated  (Art.  39)  that  the  spectroscope 
enables  us  to  establish  the  existence  of  certain  chemical 


GENERAL  DESCRIPTION  OF  THE  SUN  103 

substances  in  the  sun,  by  a  comparison  of  the  spectra  of  these 
substances  with  that  of  the  sun;  or,  more  precisely,  by  a 
comparison  of  the  lines  bright  or  dark,  by  which  these 
different  spectra  are  distinguished.  The  number  of  the 
dark  lines  in  the  solar  spectrum  which  have  been  detected 
and  mapped  exceeds  20,000;  and  careful  examination  also 
shows  that  some  of  these  are  double.  Some  of  the  more 
prominent  of  these  lines  have  received  the  names  of  the  first 
letters  of  the  alphabet;  D,  for  example,  is  a  very  noticeable 
double  line  in  the  orange  of  the  spectrum.  When  certain 
chemical  substances  are  vaporized,  either  in  a  flame  or  by 
the  electric  current,  the  spectra  which  they  form  are  also 
characterized  by  lines,  which,  however,  are  not  dark,  but 
bright.  If,  for  instance,  sodium  is  introduced  into  a  flame,  its 
incandescent  vapor  produces  a  spectrum  which  is  character- 
ized by  a  brilliant  double  line  of  yellow;  and  it  is  especially 
noticeable  that  this  yellow  line  coincides  exactly  in  position 
with  the  dark  line  D  of  the  solar  spectrum.  In  the  same  way 
the  spectrum  of  zinc  is  found  to  contain  lines  of  red  and 
blue;  that  of  copper  contains  lines  of  green;  and,  in 
general,  the  spectrum  of  each  metal  contains  certain 
bright  lines,  peculiar  to  itself,  and  readily  recognized. 
We  may  therefore  conclude  that  an  incandescent  gas  or 
vapor  emits  rays  of  a  certain  refrangibility  and  color,  and 
those  rays  only. 

Again,  it  is  proved  by  experiment  that  if  a  ray  of  white 
light  be  allowed  to  pass  through  an  incandescent  vapor,  the 
vapor  will  absorb  precisely  those  rays  which  it  can  itself  emit. 
If,  for  instance,  a  continuous  spectrum  be  formed  by  a  ray  of 
intense  white  light  from  any  source,  and  if  the  vapor  of 
sodium  be  introduced  in  the  path  of  this  ray,  between  the 
prism  and  the  source  of  light,  a  dark  line  will  appear  in 
the  spectrum,  identical  in  position  with  the  bright  yellow 
line  which  we  have  already  noticed  in  the  spectrum  of 
sodium,  and  which  we  found  to  be  identical  in  position  with 
the  dark  line  D  of  the  solar  spectrum. 


104    THE  SUN.     THE  EARTH'S  ORBIT.     THE  SEASONS 

We  are  now  ready  to  apply  the  principles  established  by 
these  experiments  to  the  case  of  the  sun.  The  sun  is,  as  we 
saw  above,  a  sphere  surrounded  by  a  vaporous  envelope. 
This  sphere  would  of  itself  emit  all  kinds  of  rays,  and  there- 
fore give  a  continuous  spectrum ;  but  the  chromosphere  which 
surrounds  it  absorbs  those  of  the  sun's  rays  which  it  can 
itself  emit.  The  dark  line  D  of  the  solar  spectrum  shows, 
as  in  the  experiment  above  described,  that  sodium  has  been 
introduced  in  the  path  of  the  sun's  rays :  in  other  words,  that 
sodium  is  in  the  sun's  chromosphere.  In  the  same  way,  Pro- 
fessor Kirchhoff,  to  whom  we  owe  this  remarkable  discovery, 
has  established  the  existence  in  the  chromosphere  of  iron, 
calcium,  magnesium,  chromium,  and  other  metals.  In  the 
case  of  iron,  more  than  450  bright  lines  have  been  detected  in 
its  spectrum:  and  for  every  one  of  these  lines  there  is  a 
corresponding  dark  line  in  the  solar  spectrum. 

We  also  see,  from  the  preceding  experiments,  how  the 
presence  of  bright  lines  in  the  spectrum  of  the  rose-colored 
protuberances  could  prove  to  Dr.  Jannsen  that  these  pro- 
tuberances were  not  masses  of  clouds,  reflecting  the  light  of 
the  sun,  but  masses  of  incandescent  vapor.  We  shall  see 
another  instance  of  the  same  description  when  we  come  to 
examine  some  of  the  nebulae. 

THE  ZODIACAL  LIGHT 

103.  At  certain  seasons  of  the  year  a  faint  nebulous  light, 
not  unlike  the  tail  of  a  comet,  is  seen  in  the  west  after  twilight 
has  ended,  or  in  the  east  before  it  has  begun.  This  is  called 
the  Zodiacal  Light.  Its  general  shape  is  nearly  that  of  a  cone, 
the  base  of  which  is  turned  towards  the  sun.  The  breadth 
of  the  base  varies  from  8°  to  30°  of  angular  magnitude.  The 
apex  of  the  cone  is  sometimes  more  then  90°  to  the  rear  or  in 
advance  of  the  sun.  According  to  Humboldt,  it  is  almost 
always  visible,  at  the  times  above  stated,  within  the  tropics : 
in  our  latitudes  it  is  seen  to  the  best  advantage  in  the  evening 


THE  ZODIACAL  LIGHT  105 

near  the  first  of  March,  and  in  the  morning  near  the  middle 
of  October. 

Of  the  many  theories  proposed  to  account  for  the  zodiacal 
light,  the  one  which  seems  to  be  most  widely  accepted  is 
that  it  consists  of  a  ring  or  zone  of  rare  nebulous  matter 
encircling  the  sun,  which  reaches  as  far  as  the  earth,  and 
perhaps  extends  beyond  it.  According  to  another  theory, 
it  is  a  belt  of  meteoric  bodies  surrounding  the  sun.  A  very 
interesting  and  valuable  series  of  observations  upon  the 
Zodiacal  Light  was  made  by  Chaplain  Jones,  United  States 
Navy,  in  the  years  1853-5,  in  latitudes  ranging  from  41°  N. 
to  53°  S.  The  conclusion  which  he  drew  from  his  observa- 
tions was  that  the  light  was  a  nebulous  ring  encircling  the 
earth,  and  lying  within  the  orbit  of  the  moon. 


CHAPTER  VII 

SIDEREAL  AND  SOLAR  TIME.     THE  EQUATION  OF  TIME. 
THE  CALENDAR 

104.  Sidereal  and  Solar  Days. — It  is  important  to  distin- 
guish between  the  apparent  annual  motion  of  the  sun  in  the 
ecliptic,  from  west  to  east,  and  the  apparent  diurnal  motion 
towards  the  west,  which  the  rotation  of  the  earth  gives  to  all 
celestial  bodies  and  points.     A  sidereal  day  is  the  interval  of 
time  between  two  successive  transits  of  the  vernal  equinox 
over  the  same  branch  of  the  meridian.     A  solar  day  is  the 
interval  between  two  similar  transits  of  the  sun.     But  the 
continuous  motion  of  the  sun  towards  the  east  causes  it  to 
appear  to  move  more  slowly  towards  the  west  than  the 
vernal  equinox  moves.     The  solar  day  is  therefore  longer  than 
the  sidereal  day,  the  average  amount  of  the  difference  being 
3m.  55.9s.     And  furthermore,  in  the  interval  of  time  in 
which  the  sun  makes  one  complete  revolution  in  the  ecliptic, 
the  number  of  daily  revolutions  which  it  appears  to  make 
about  the  earth  will  be  less  by  one  than  the  number  of  daily 
revolutions  made  by  the  equinox.     The  sidereal  year,  then 
(Art.  89),  which  contains  365d.  6h.  9m.  9.5s.  of  solar  time, 
contains  366d.  6h.  9m.  9.5s.  of  sidereal  time. 

105.  Relation  of  Sidereal  and  Solar  Times. — Since  the 
sidereal  day  is  shorter  than  the  solar  day  (and,  consequently, 
the  sidereal  hour,  minute,  etc.,  than  the  solar  hour,  minute, 
etc.),  it  is  evident  that  any  given  interval  of  time  will  contain 
more  units  of  sidereal   than  of  solar  time.     The  relative 

106 


THE  EQUATION  OF  TIME  107 

values  of  the  sidereal  and  the  solar  days,  hours,  etc.,  are 
obtained  as  follows:  We  have  from  the  preceding  article, 

366.25636  sidereal  days  =  365.25636  solar  days; 
one  sidereal  day  =      0.99727  solar  day; 
one  sidereal  hour  =      0.99727  solar  hour,  etc. 

Having,  therefore,  an  interval  of  time  expressed  in  either  solar 
or  sidereal  units,  we  may  easily  express  the  same  interval  in 
units  of  the  other  denomination.  This  is  called  the  conver- 
sion of  a  solar  into  a  sidereal  interval,  and  the  reverse:  and 
tables  for  facilitating  this  conversion  are  given  in  the  Nautical 
Almanac. 

Again,  knowing  the  sidereal  time  at  any  instant,  the  hour- 
angle,  that  is  to  say,  of  the  vernal  equinox,  the  corresponding 
solar  time,  or  the  hour-angle  of  the  sun,  is  readily  obtained  by 
subtracting  from  the  sidereal  time  the  sun's  right  ascension. 
This  is  indeed  a  corollary  of  the  theorem  proved  in  Art.  9, 
from  which  we  see  that  the  sum  of  the  sun's  right  ascension 
(which  can  always  be  found  in  the  Nautical  Almanac),  and 
its  hour-angle,  is  the  sidereal  time.  Either  of  these  times, 
then,  may  be  converted  into  the  other. 

THE  EQUATION  OF  TIME 

106.  Inequality  of  Solar  Days. — Observation  shows  that 
the  length  of  the  solar  day  is  not  a  constant  quantity,  but 
varies  at  different  seasons  of  the  year,  and,  indeed,  from  day 
to  day.  A  distinction  must  therefore  be  made  between  the 
apparent  or  actual  solar  day,  and  the  mean  solar  day,  which  is 
the  mean  of  all  the  apparent  solar  days  of  the  year.  A 
uniform  measure  of  time  may  be  obtained  from  the  apparent 
diurnal  motion  with  reference  to  our  meridian  of  a  fixed 
celestial  body  or  point.  It  may  also  be  obtained  from  the 
apparent  diurnal  motion  of  a  celestial  body  which  changes 
its  position  in  the  heavens,  provided  that  two  conditions  are 
satisfied;  first,  the  plane  in  which  the  body  moves  must  be 


108  SIDEREAL  AND  SOLAR  DAYS 

perpendicular  to  the  plane  of  the  meridian:  and  second,  its 
motion  in  that  plane  must  be  uniform.  Both  these  condi- 
tions are  so  very  nearly  satisfied  by  the  motion  of  the  vernal 
equinox  that  any  two  sidereal  days  may  be  considered  to  be 
sensibly  equal  to  each  other;  but  neither  condition  is  satisfied 
by  the  motion  of  the  sun.  It  moves  in  the  ecliptic,  the  plane 
of  which  is  not,  in  general,  perpendicular  to  the  plane  of  the 
meridian:  and  its  motion  in  this  plane  is  not  uniform.  We 
have,  therefore,  two  causes  of  the  inequality  of  the  solar  days, 
the  effect  of  each  of  which  we  will  now  proceed  to  examine. 
107.  Irregular  Advance  of  the  Sun  in  the  Ecliptic. — Ob- 
servation shows  that  the  sun's  motion  in  longitude  is  not 
uniform.  The  mean  daily  motion  is, 
of  course,  obtained  by  dividing  360° 
by  the  number  of  days  and  parts  of  a 
day  in  a  year,  and  is  59'  8".2.  But 
the  daily  motion  about  the  first  of 
January  is  61'  10",  while  about  the 
first  of  July  it  is  only  57'  12".  In  Fig. 
42,  let  the  circle  AM'  M"  represent 
FIG.  42.  fae  apparent  orbit  of  the  sun  in  the 

ecliptic  about  the  earth   E,  and  let 

the  sun  be  supposed  to  be  at  the  point  A  where  its  daily 
motion  is  the  greatest,  on  the  first  of  January.  Let  us 
also  suppose  a  fictitious  sun  (which  we  will  call  the  first 
mean  sun)  to  move  in  the  ecliptic  with  the  uniform  rate  of 
59'  8".2  daily,  and  to  be  at  the  point  A  at  the  same  time 
that  the  true  sun  is  there.  On  the  next  day  the  first  mean 
sun  will  have  moved  eastward  to  some  point  M,  while  the 
true  sun,  whose  daily  motion  is  at  this  time  greater  than  that 
of  the  first  mean  sun,  will  be  found  at  some  point  T,  to  the 
east  of  M.  The  true  sun  will  continue  to-  gain  on  this  mean 
sun  for  about  three  months,  at  the  end  of  which  time  this 
mean  sun  will  begin  to  gain  on  the  true  sun,  and  will  finally 
overtake  it  at  the  point  B,  on  the  first  of  July.  During  the 
second  half  of  the  year  the  first  mean  sun  will  be  to  the  east  of 


THE  EQUATION  OF  TIME 


109 


the  true  sun,  and  at  the  end  of  the  year  the  two  suns  will 
again  be  together  at  A. 

The  angular  distance  between  the  two  suns,  represented 
in  the  figure  by  the  angles  TEM,  T'EM',  etc.,  is  called  the 
Equation  of  the  Center.  It  is  evidently  additive  to  the  mean 
longitude  of  the  sun  while  it  is  moving  from  A  to  B,  and 
subtractive  from  it  while  it  is  moving  from  B  to  A.  Its 
greatest  value  is  about  8  minutes  of  time. 

Since  the  rotation  of  the  earth  gives  to  both  these  bodies  a 
common  daily  motion  to  the  west,  it  is  plain  that  from  Jan- 
uary to  July  the  first  mean  sun  will  cross  the  meridian  before 
the  true  sun,  and  that  from  July  to  January  the  true  sun  will 
cross  the  meridian  before  the  first  mean  sun. 

108.  Obliquity  of  the  Ecliptic  to  the  Meridian. — Even  if 
the  sun's  motion  in  the  ecliptic  were  uniform,  equal  advances 
of  the  sun  in  longitude  would  not  be  accompanied  by  equal 
advances  in  right  ascension,  in  consequence  of  the  obliquity 
of  the  ecliptic  to  the  meridian.  The  truth  of  this  may  be 
seen  in  Fig.  43.  Let  this  figure  represent  the  projection  of 
the  celestial  sphere  on  the  plane 
of  the  equinoctial  colure  PApH. 
A  and  H  are  the  equinoxes.  P 
and  p  the  celestial  poles,  AeH 
the  equinoctial,  and  AEH  the 
ecliptic.  Let  the  ecliptic  be 
divided  into  equal  arcs,  A  B,  BC, 
etc.,  and  through  the  points  of 
division,  B,  C,  etc.,  let  hour- 
circles  be  drawn,  meeting  the 
equinoctial  in  the  points  6,  c, 
etc.  Now,  since  all  great  circles 

bisect  each  other,  AEH  is  equal  to  AeH,  and  if  Pep  is 
the  projection  of  an  hour-circle  perpendicular  to  the  circle 
PApH,  AE  and  Ae  are  quadrants,  and  equal.  The  angle 
PBC  is  evidently  greater  than  PAB,  PCD  is  greater  than 
PBC,  etc.:  in  other  words,  the  equal  arcs  AB}  BC,  etc., 


FIG.  43. 


110  SIDEREAL  AND  SOLAR  DAYS 

are  differently  inclined  to  the  equinoctial.  The  effect  of  this 
is  that  the  equinoctial  is  divided  into  unequal  parts  by  the 
hour-circles  Pb,  PC,  etc.,  be  being  greater  than  Ab,  cd  than  be, 
etc.  It  is  to  be  noticed,  further,  that  the  points  B  and  b, 
being  on  the  same  hour-circle,  will  be  on  the  meridian  at 
the  same  instant  of  time :  and  the  same  is  true  of  C  and  c,  D 
and  d,  etc. 

Now,  if  A  is  the  vernal  equinox,  the  first  mean  sun, 
moving  in  the  ecliptic  with  the  constant  daily  rate  of  59' 
8".2,  will  pass  through  that  point  on  the  21st  of  March.  Let 
another  fictitious  sun  (called  the  second  mean  sun)  leave  the 
point  A  at  the  same  time,  and  move  in  the  equinoctial  with 
the  same  uniform  daily  rate.  Since  BAb  is  a  right-angled 
triangle,  Ab  is  less  than  A  B.  Hence,  when  the  first  mean 
sun  reaches  B,  the  second  mean  sun  will  be  at  some  point 
m,  to  the  east  of  b :  when  the  first  mean  sun  is'  at  C,  the  second 
mean  sun  will  be  to  the  east  of  c,  etc.,  and  the  second  mean 
sun  will  continue  to  lie  to  the  east  of  the  first  mean  sun  until 
the  21st  of  June  (the  summer  solstice),  when  both  suns  will  be 
at  the  points  E  and  e  at  the  same  instant  of  time,  and  will 
therefore  come  to  the  meridian  together.  In  the  second 
quadrant,  the  second  mean  sun  will  lie  to  the  west  of  the  first 
mean  sun,  and  both  suns  will  reach  H,  the  autumnal  equinox, 
on  the  21st  of  September.  The  relative  positions  in  the  third 
and  the  fourth  quadrant  will  be  identical  with  those  in  the 
first  and  the  second. 

From  the  21st  of  March,  then,  to  the  21st  of  June,  the 
second  mean  sun,  being  to  the  east  of  the  first  mean  sun,  will 
come  later  to  the  meridian;  and  the  same  will  also  be  true  from 
the  21st  of  September  to  the  21st  of  December.  In  the  two 
other  similar  periods  the  case  will  be  reversed,  and  the  second 
mean  sun  will  come  earlier  to  the  meridian  than  the  first 
mean  sun.  The  greatest  difference  of  the  hour-angles  of 
these  two  mean  suns  is  about  10  minutes  of  time. 

109.  Equation  of  Time. — It  is  by  means  of  these  two 
fictitious  suns  that  we  are  able  to  obtain  a  uniform  measure  of 


THE  EQUATION  OF  TIME 


111 


time  from  the  irregular  advance  of  the  sun  in  the  ecliptic. 
The  second  mean  sun  satisfies  the  two  conditions  stated  in 
Art.  106,  and  therefore  its  hour-angle  is  perfectly  uniform  in 
its  increase.  This  hour-angle  is  the  mean  solar  time  of  our 
ordinary  watches  and  clocks.  The  hour-angle  of  the  true 
sun  is  called  the  apparent  solar  time:  and  the  difference  at 
any  instant  between  the  apparent  and  the  mean  solar  time  is 
called  the  equation  of  time.  Let  Fig.  44  be  a  projection  of  the 
celestial  sphere  on  the  plane  of  the  horizon.  Z  is  the  zenith, 
P  the  pole,  EVQ  the  equinoctial,  CL  the  ecliptic,  and  V 
the  vernal  equinox.  Let  T  be 
the  position  of  the  true  sun 
in  the  ecliptic,  and  M  that  of 
second  mean  sun  in  the  equi- 
noctial. The  angle  TPM  is 
evidently  the  equation  of  time. 
This  angle  is  measured  by 
the  arc  AM,  or  VM—VA:  the 
difference,  that  is,  of  the  right 
ascensions  of  the  true  and  the 
second  mean  sun.  But  since 
the  angular  advance  of  the 

second  mean  sun  in  the  equinoctial  is,  by  hypothesis,  as 
shown  in  the  previous  article,  equal  to  the  angular  advance  of 
the  first  mean  sun  in  the  ecliptic,  it  follows  that  the  right 
ascension  of  the  second  mean  sun  is  always  equal  to  the 
longitude  of  the  first  mean  sun,  or,  as  it  is  usually  called,  the 
true  sun's  mean  longitude.  The  equation  of  time,  then,  is 
the  difference  of  the  sun's  true  right  ascension  and  mean  longi- 
tude', and  thus  computed  is  given  in  the  Nautical  Almanac 
for  every  two  hours  of  each  day  in  the  year.  It  reduces  to 
zero  four  times  in  the  year,  and  passes  through  four  maxima, 
ranging  in  value  from  4  minutes  to  16  minutes. 

110.  Astronomical  and  Civil  Time. — The  mean  solar  day 
is  considered  by  astronomers  to  begin  at  mean  noon,  when 
the  second  mean  sun  (usually  called  simply  the  mean  sun) 


FIG.  44. 


112  SIDEREAL  AND  SOLAR  DAYS 

is  at  its  upper  culmination.  The  hours  are  reckoned  from 
Oh.  to  24th.  The  mean  solar  day,  so  considered,  is  called 
the  astronomical  day. 

The  civil  day  begins  at  midnight,  twelve  hours  before  the 
astronomical  day,  and  is  divided  into  two  parts  of  twelve 
hours  each,  called  A.M.  and  P.M. 

We  must,  therefore,  carefully  distinguish  between  any 
given  civil  time  and  the  corresponding  astronomical  time. 
For  instance,  January  3d,  8  A.M.,  in  civil  time,  is  the  same  as 
January  2d,  20h.,  in  astronomical  time. 

THE  CALENDAR 

111.  Owing  to  causes  which  will  be  explained  further  on, 
the  position  of  the  vernal  equinox  is  not  absolutely  stationary, 
but  moves  westward  along  the  ecliptic,  with  an  annual  rate  of 
about  50".26.  The  sun,  then,  moving  eastward  from  the 
equinox,  will  reach  it  again  before  it  has  made  one  complete 
sidereal  revolution  about  the  earth.  This  interval  of  time  in 
which  the  sun  moves  from  and  returns  to  the  equinox  is 
called  a  tropical  year,  and  consists  of  365d.  5h.  48m.  45.9s. 
The  Julian  Calendar  was  established  by  Julius  Caesar,  44 
B.C.,  and  by  it  one  day  was  inserted  in  every  fourth  year. 
This  was  the  same  thing  as  assuming  that  the  length  of  the 
tropical  year  was  365d.  6h.,  instead  of  the  value  given  above, 
thus  introducing  an  accumulative  error  of  llm.  14s.  every 
year.  This  calendar  was  adopted  by  the  Church  in  A.D., 
325  at  the  Council  of  Nice,  and  the  vernal  equinox  then 
fell  on  the  21st  of  March.  In  1582  the  annual  error  of  llm. 
14s.  caused  the  vernal  equinox  to  fall  on  the  llth  of  March, 
instead  of  the  21st.  Pope  Gregory  XIII.  therefore  ordered 
that  ten  days  should  be  omitted  from  the  year  1582,  and  thus 
brought  the  vernal  equinox  back  again  to  the  21st  of  March. 
Furthermore,  since  the  error  of  llm.  14s.  a  year  amounted 
to  very  nearly  three  days  in  400  years,  it  was  decided  to 
leave  out  three  of  the  inserted  days  (called  intercalary  days) 


THE  CALENDAR  113 

every  400  years,  and  to  make  this  omission  in  those  years 
which  were  not  exactly  divisible  by  400.  Thus  of  the  years 
1700,  1800,  1900,  2000,  all  of  which  are  leap  years  according 
to  the  Julian  calendar,  only  the  last  is  a  leap  year  according 
to  the  reformed  or  Gregorian  calendar.  By  this  calendar  the 
annual  error  is  only  24  seconds,  and  will  not  amount  to  a  day 
in  much  less  than  4000  years. 

This  reformed  calendar  was  not  adopted  by  England  until 
1752,  when  eleven  days  were  omitted  from  the  calendar. 
The  two  calendars  are  now  often  called  the  old  style  and  the 
new  style.  For  instance,  April  26th,  O.S.,  is  the  same  as 
May  9th,  N.S.  In  Russia  the  old  style  is  still  retained, 

though  it  is  customary  to  give  both  dates;  as  1920  -=-^- 

Feb.  5 

All  other  Christian  countries  have  adopted  the  new  style. 

It  may  be  noted,  as  a  matter  of  interest,  that  the  time  of 
beginning  the  year  has  varied  greatly  in  different  countries 
and  in  different  centuries.  The  Athenian  year  began  in 
June,  and  the  Persian  year  in  August.  Holden  says  (Ele- 
mentary Astronomy):  "The  most  common  times  of  com- 
mencing were,  perhaps,  March  1  and  March  22.  But  Jan- 
uary 1  gradually  made  its  way,  an)d  became  universal  after 
its  adoption  by  England  in  1752."  See  The  American 
Ephemeris,  page  xv. 


CHAPTER  VIII 

LAW  OF  UNIVERSAL  GRAVITATION.     PERTURBATIONS  IN 
THE  EARTH  S  ORBIT.     ABERRATION 

112.  The  Law  of  Universal  Gravitation.— The  earth,  as 
we  have  seen  in  Chapter  VI.,  revolves  about  the  sun  in  an 
elliptical  orbit,  with  a  linear  velocity  of  eighteen  miles  a 
second.  At  every  point  of  its  orbit  the  centrifugal  force 
induced  by  this  revolution  must  create  in  the  earth  a  ten- 
dency to  leave  its  orbit,  and  to  go  off  in  the  direction  of  a 
tangent  to  the  orbit  at  that  point.  To  counteract  this 
centrifugal  force,  there  must  constantly  exist  a  centripetal 
force,  by  which  the  earth  is  at  every  instant  deflected  from 
this  rectilinear  path  which  it  tends  to  follow,  and  is  drawn 
towards  the  sun ;  and  in  order  that  the  orbit  of  the  earth  may 
remain  unchanged  in  form, — as  observation  shows  that  it 
does  remain, — these  two  forces  must  be  in  constant  equilib- 
rium. Admitting,  then,  the  existence  of  such  a  centripetal 
force,  it  remains  to  determine  the  nature  of  the  force,  and 
the  laws  under  which  it  acts. 

The  force  is  believed  to  be  identical  in  nature  with  that 
force  which  causes  all  bodies,  free  to  move,  to  tend  towards 
the  earth's  center,  and  which  we  call  the  force  of  gravity. 
At  whatever  height  above  the  surface  of  the  earth  the  ex- 
periment may  be  made,  this  attractive  force  of  the  earth  is 
found  to  exist;  and  there  is  no  good  reason  for  assuming  any 
finite  limit  beyond  which  this  force,  however  much  its  effects 
may  be  lessened  by  other  and  opposing  forces,  does  not  have 
at  least  a  theoretic  existence.  And,  furthermore,  as  the 
sun  and  the  other  heavenly  bodies  are  all  masses  of  matter 

114 


THE  LAW  OF  UNIVERSAL  GRAVITATION  115 

like  the  earth,  there  is  every  reason  for  concluding  that  they 
too,  as  well  as  the  earth,  possess  this  power  of  attracting 
other  bodies  towards  their  centers.  Nor  is  this  attractive 
power  a  characteristic  of  large  bodies  alone:  for  we  have 
already  seen  in  the  experiment  with  the  torsion  balance 
described  in  Art.  66,  that  small  globes  of  lead  exert  a  sensible 
attraction  upon  still  smaller  globes.  We  may  therefore 
assume  that  what  is  true  of  each  of  these  masses,  large  and 
small,  as  a  whole,  is  no  less  true  of  the  separate  particles  of 
which  it  is  composed,  and  that  every  particle  of  matter 
in  the  universe  has  an  attractive  power  upon  every  other 
particle. 

In  order  to  determine  the  laws  under  which  this  attractive 
power  is  exerted,  we  have  only  to  assume  that  the  laws  which 
are  shown  by  experiment  to  obtain  at  the  earth's  surface  hold 
equally  good  throughout  the  universe;  so  that  whatever  the 
masses  of  bodies  may  be,  or  whatever  the  distances  by  which 
they  are  separated  from  each  other,  the  forces  with  which  any 
two  bodies  attract  a  third  will  be  directly  proportional  to 
the  masses  of  the  two  attracting  bodies,  and  inversely  pro- 
portional to  the  squares  of  their  distances  from  the  third 
body. 

This,  then,  is  Newton's  Law  of  Universal  Gravitation. 
Every  particle  of  matter  in  the  universe  attracts  every  other 
particle,  with  a  force  directly  proportional  to  the  mass  of  the 
attracting  particle,  and  inversely  proportional  to  the  square 
of  the  distance  between  the  particles.  In  applying  this  general 
law  to  the  particles  which  compose  the  masses  of  the  heavenly 
bodies,  Newton  has  demonstrated  that  the  attraction  exerted 
by  a  sphere  is  precisely  what  it  would  be  if  all  the  particles 
in  the  sphere  were  collected  at  its  center,  and  constituted  one 
particle,  with  an  attractive  power  equal  to  the  sum  of  the 
powers  of  these  different  particles. 

113.  Verification  of  the  Law  in  the  Case  of  the  Moon. — 
The  moon  is  shown  by  observation  to  revolve  about  the 
earth  in  a  period  of  27.32  days,  at  a  mean,  distance  from  the 


116  LAW  OF  UNIVERSAL  GRAVITATION 

earth  of  238,862  miles.     If  we  take  the  formula  for  centrif- 
ugal force  given  in  Art.  69, 


and  substitute  for  r  the  moon  s  distance  in  feet,  and  for  t  its 
period  of  revolution  in  seconds,  m  the  mass,  we  shall  find  for 
the  centrifugal  force, 

/=  0.0089  X  mass  of  the  moon: 

that  is  to  say,  in  one  second  the  earth  tends  to  give  the  moon 
a  velocity  towards  itself  of  0.0089.  Now  the  force  of 
gravity  on  the  earth's  surface  at  the  equator  is  32.09  pounds; 
and  if  the  law  of  gravitation  is  assumed  to  be  true,  the  force 

on  rvq 

of  gravity  at  the  distance  of  the  moon  will  be  .  Q  '     >2 

pounds,  since  the  distance  of  the  moon  from  the  earth  is 
equal  to  60.267  of  the  earth's  radii.  The  value  of  this 
expression  is  found  to  be  0.0088.  The  two  results  vary  by 
only  TWO^:  and  it  is  therefore  fair  to  conclude  that  the 
centrifugal  force  of  the  moon  in  its  orbit  is  really  counter- 
acted by  the  earth's  attraction. 

In  whatever  way  the  law  of  gravitation  is  tested  in  con- 
nection with  the  observed  motions  of  the  heavenly  bodies, 
the  facts  which  come  by  observation  are  always  found  to  be 
in  close  agreement  with  the  results  which  the  law  demands; 
and  it  is  safe  to  say  that  the  truth  of  this  law  is  as  satisfacto- 
rily demonstrated  as  is  that  of  the  laws  of  refraction,  of  the 
laws  of  sound,  or  of  the  many  other  natural  laws  which 
depend  upon  observation  and  experiment  for  their  ultimate 
proof. 

114.  The  Mass  of  the  Sun.  —  Let  A  denote  the  attraction 
exerted  by  the  sun  on  the  earth,  and  a  that  exerted  by  the 
earth  on  a  body  at  its  surface.  Let  M  denote  the  mass  of  the 
sun,  m  that  of  the  earth  ,  r  the  radius  of  the  earth,  and  R  the 


THE  MASS  OF  THE  SUN  117 

radius  of  the  earth's  orbit.     We  have,  then,  by  the  law  of 
gravitation, 

A    M    & 

~~~  /\    7~*O* 

a    m     Rz 
But  A  must  equal  the  earth's  centrifugal  acceleration  in  its 

A     2  D 

orbit,  or  -^y-,   in   which   t   is  365.256   days,   reduced   to 

seconds,  and  R  is  expressed  in  feet.     We  have  also  a  equal 
to  32.09  feet.     Substituting  these  values,  we  have, 

M      4ir2R3 


m     32.09ZV 

Substituting  the  known  values  of  the  different  quantities,  we 
shall  have 

—  =  327,000: 
m 

or  the  mass  of  the  sun  is  equal  to  that  of  327,000  earths. 
The  density  of  the  sun,  compared  with  that  of  the  earth, 

327  000 

being  equal  to  the  mass  divided  by  the  volume,  is         ' . 

1,300,000 

The  density  is  therefore  equal  to  about  Jth  of  that  of  the 
earth,  and  to  about  |ths  of  that  of  water. 

115.  Weight  of  Bodies  at  the  Surface  of  the  Sun. — The 
weight  of  the  same  body  at  the  surface  of  the  earth  and  at 
that  of  the  sun  will  be  directly  as  the  masses  of  the  two 
spheres  and  inversely  as  the  squares  of  their  radii.     We  shall 
find  that  the  weight  of  a  body  at  the  sun  is  about  27.7  times 
its  weight  at  the  earth ;  so  that  a  body  which  exerts  a  pressure 
of  10  pounds  at  the  earth  would  exert  a  pressure  at  the  sun 
equal  to  that  of  277  of  the  same  pounds;  and  a  man  whose 
weight  is  150  pounds  would,  if  transported  to  the  sun,  be 
obliged  to  support  in  his  own  body  a  weight  equivalent  to 
about  two  of  our  tons. 

116.  The  Earth's  Motion  at  Perihelion  and  Aphelion.— 
We  have  already  seen  that  the  angular  velocity  of  the  earth 


118  LAW  OF  UNIVERSAL  GRAVITATION 

in  its  orbit  is  the  greatest  at  perihelion,  when  the  earth  is  the 
nearest  to  the  sun, .and  is  the  least  at  aphelion,  when  the 
earth  is  the  farthest  from  the  sun.  This  irregularity  of 
motion  is  a  consequence  of  the  attraction  exerted  by  the 
sun  on  the  earth,  as  may  be  seen  in  Fig.  45.  In  this  figure, 
let  S  be  the  sun,  P  the  perihelion  of  the  earth's  orbit,  and  A 

the  aphelion.  Let  the  earth 
move  from  A  to  P,  and  suppose 
it  to  be  at  the  point  E.  The 
attraction  of  the  sun  on  the 
earth,  along  the  line  ES,  may 
be  resolved  into  two  forces,  one 
of  which,  acting  in  the  direc- 
tion of  the  tangent  EB,  will 
evidently  tend  to  increase  the 
velocity  of  the  earth  in  its 
FIG.  45.  orbit.  At  the  point  E",  on  the 

contrary,    where    the    earth    is 

moving  toward  A,  the  effect  of  the  sun's  attraction  is  to 
diminish  the  earth's  velocity.  In  general,  then,  the  earth's 
velocity  will  increase  as  it  moves  from  aphelion  to  perihelion, 
and  decrease  as  it  moves  from  perihelion  to  aphelion. 

KEPLER'S  LAWS 

117.  In  the  early  part  of  the  seventeenth  century,  more 
than  fifty  years  before  the  announcement  by  Newton  of  the 
law  of  universal  gravitation,  the  astronomer  Kepler,  by  an 
examination  of  the  observations  which  had  been  made  upon 
the  motions  of  the  planets,  and  which  had  shown  that  the 
planets  revolved  about  the  sun,  discovered  the  following 
laws: 

(1)  The  orbit  of  every  plant  is  an  ellipse,  having  the  sun 
at  one  of  its  foci. 

(2)  If  a  line,  called  a  radius  vector,  is  supposed  to  be  drawn 
from  the  sun  to  any  planet,  the  areas  described  by  this  line,  as 
the  planet  revolves  in  its  orbit,  are  proportional  to  the  times. 


KEPLER'S  LAWS  119 

(3)  The  squares  of  the  times  of  revolution  of  any  two 
planets  are  proportional  to  the  cubes  of  their  mean  distances 
from  the  sun. 

These  laws  were  verified  by  Newton  in  his  Principia,  in  a 
course  of  mathematical  reasoning,  the  foundation  of  which 
was  his  theory  of  universal  gravitation.  With  regard  to 
Kepler's  first  law,  he  showed  that  any  two  spherical  bodies, 
mutually  attracted,  describe  orbits  about  their  common 
center  of  gravity,  and  that  these  orbits  are  limited  in  form  to 
one  or  another  of  the  four  conic  sections, — the  circle,  the 
ellipse,  the  parabola,  and  the  hyperbola.  For  instance,  in 
the  case  of  the  earth  and  the  sun,  the  earth  does  not  describe 
an  ellipse  about  the  sun  at  rest,  but  both  earth  and  sun  re- 
volve about  their  common  center  of  gravity.  It  is  shown  in 
mechanics  that  the  common  center  of  gravity  of  any  two 
globes  is  at  a  point  on  the  straight  line  joining  their  inde- 
pendent centers  of  gravity,  so  situated  that  its  distances  from 
the  centers  of  the  two  globes  are  inversely  as  the  masses  of 
the  globes.  Hence  the  distance  of  the  common  center  of 
gravity  of  the  earth  and  the  sun  from  the  center  of  the  sun  is 

92  897  000 

'       '        miles,  or  only  about  282  miles;  so  that  the  sun 

oZi  ,UUU 
may  practically  be  considered  to  be  at  rest. 

With  regard  to  Kepler's  second  law,  Newton  further 
showed  that  the  angular  velocity  with  which  the  radius  vector 
moves  must  be  inversely  proportional  to  the  square  of  its 
length.  Finally,  he  proved  also  the  truth  of  the  third  law, 
provided  only  that  a  slight  correction  is  introduced  when  the 
mass  of  the  planet  is  not  so  small  as  to  be  inappreciable  in 
comparison  with  that  of  the  sun.  If  t  and  t'  denote  the  times 
of  revolution  of  any  two  planets,  m  and  m'  their  masses 
(the  mass  of  the  sun  being  unity),  and  d  and  df  their  mean 
distances  from  the  sun.  Newton  showed  that  we  shall  al- 
ways have  the  following  proportion: 

,73          ,7/3 
t2  .  t>2=_^_  .     a 

\-\-m     \-\-m 


120 


LAW  OF  UNIVERSAL  GRAVITATION 


If  m  and  mf  are  so  small  that  they  may  be  omitted  with- 
out sensible  error,  this  proportion  becomes  identical  with 
Kepler's  third  law. 


PERTURBATIONS  IN  THE  EARTH'S  ORBIT 

118.  Precession. — Although  the  absolute  positions  of  the 
planes  of  the  ecliptic  and  the  equinoctial  in  space,  and  their 
relative  positions  to  each  other,  remain  very  nearly  the  same 
from  year  to  year,  there  are  nevertheless  certain  small  per- 
turbations in  these  positions  which  are  made  evident  to  us  by 
refined  and  extended  observations.  The  principal  of  these 
perturbations  is  called  precession. 

The  latitudes  of  all  the  fixed  stars  remain  very  nearly  the 
same  from  year  to  year,  and  even  from  century  to  century: 
and  we  therefore  conclude  that  the  position  of  the  ecliptic 
with  reference  to  the  celestial  sphere  remains  very  nearly 
unchanged.  But  the  longitudes  of  the  stars  are  all  found  to 
increase  by  an  annual  amount  of  50".26 :  and  hence  the  line 
of  the  equinoxes  must  have  an  annual  westward  motion  of 
the  same  amount.  This  westward  motion  is  called  the 
precession  of  the  equinoxes.  Since  the  ecliptic  remains 
stationary  in  the  heavens  (or  at  least  so  nearly  stationary 

that  the  latitudes  of  the  stars 
only  vary  by  half  a  second  of 
arc  in  a  year),  this  precession 
must  be  considered  to  be  a 
motion  of  the  equinoctial  on  the 
ecliptic. 

In  Fig.  46,  let  EQ  repre- 
sent the  equinoctial,  and  LC  the 
ecliptic.  BV  is  the  line  of  the 
equinoxes,  which  moves  about 
in  the  plane  of  the  ecliptic,  tak- 
ing in  course  of  time  the  new 
Perhaps  the  clearest  conception  of  this 


FIG.  46. 


position   J5'7> 


PERTURBATIONS  IN  THE  EARTH'S  ORBIT          121 

motion  is  obtained  by  considering  P,  the  pole  of  the  equinoc- 
tial, to  revolve  about  A,  the  pole  of  the  ecliptic,  in  the  circle 
PG  (the  polar  radius  of  which,  AP,  is  23°  27'),  moving  west- 
ward in  this  circle  with  the  annual  rate  of  50".26,  and  com- 
pleting its  revolution  in  25,776  years. 

A  general  explanation  of  the  cause  of  precession  may  be 
given  by  means  of  Fig.  47.  The  earth  may  be  regarded  as  a 
sphere  surrounded  by  a  spheroidal  shell,  as  represented  in 


FIG.  47. 

the  figure  by  EPQp,  and  the  matter  in  this  shell  may  be 
considered  to  form  a  ring  about  the  earth  in  the  plane  of  the 
equator,  as  shown  in  E'P'Q'p'.  It  is  to  the  attraction  of  the 
sun  and  the  moon  on  this  ring,  combined  with  the  earth's 
rotation,  that  the  precession  of  the  equinoxes  is  due.  In  the 
figure  let  S  be  the  sun,  the  circle  ABV  the  ecliptic,  and 
E"Q"  this  equatorial  ring  of  the  earth.  Let  ACV  be  the 
plane  of  the  equinoctial,  meeting  the  plane  of  the  ecliptic  in 
the  line  of  equinoxes  AV.  This  plane  is  by  definition  de- 


122  LAW  OF  UNIVERSAL  GRAVITATION 

termined  at  each  instant  by  the  position  of  the  earth's 
equator.  The  attraction  exerted  by  the  sun  upon  the  dif- 
ferent particles  of  the  ring  in  that  half  of  it  which  is  nearer 
the  sun  (the  particle  E",  for  instance),  may  be  resolved  into 
two  forces,  one  acting  in  the  plane  of  the  equator,  and  the 
other  in  a  direction  perpendicular  to  that  plane,  or  in  the 
direction  E"d.  The  sun's  attraction  upon  the  nearer  half  of 
the  ring,  then,  tends  to  draw  the  plane  of  the  ring  nearer  to 
the  plane  of  the  ecliptic.  On  the  other  hand,  the  sun's  attrac- 
tion upon  the  farther  half  of  the  ring  tends  to  bring  about  the 
opposite  result;  but  since,  by  the  law  of  attraction,  the  latter 
effect  is  less  -than  the  former,  we  may  consider  the  whole 
result  of  the  sun's  attraction  upon  the  ring  to  be  a  tendency 
in  the  plane  of  the  ring  to  come  nearer  to  the  plane  of  the 
ecliptic,  i.e.,  a  rotation  about  the  line  of  equinoxes.  There- 
fore, if  the  ring  did  not  rotate,  the  plane  of  the  earth's  equator 
would  ultimately  come  into  coincidence  with  the  plane  of 
the  ecliptic. 

But  the  ring  does  rotate,  about  an  axis  perpendicular  to 
its  own  plane ;  and  the  combined  result  of  this  rotation  and  of 
the  rotation  about  the  line  of  the  equinoxes,  above  described, 
is  that  the  plane  of  the  equinoctial,  while  it  preserves  con- 
stantly its  inclination  to  the  plane  of  the  ecliptic,  moves  about 
in  a  westerly  direction:  the  line  of  intersection  of  the  two 
planes  also  moving  about  in  the  same  direction,  and  thus 
giving  rise  to  the  precession  of  the  equinoxes. 

Similar  results  will  evidently  follow  if  S  represents  the 
moon  instead  of  the  sun.  Owing  to  the  greater  proximity  of 
the  moon  to  the  earth,  however,  the  results  of  its  attraction 
are  more  than  double  those  of  the  attraction  of  the  sun. 
There  is  still  another  perturbation  in  the  position  of  the  line 
of  the  equinoxes  which  is  a  result  of  the  mutual  attraction 
between  the  earth  and  the  other  planets.  This  attraction 
tends  to  draw  the  earth  out  of  the  plane  of  the  ecliptic,  with- 
out affecting  in  any  way  the  position  of  the  plane  of  the 
,  equinoctial.  The  result  is  an  annual  movement  of  the 


PERTURBATIONS  IN  THE  EARTH'S  ORBIT          123 

equinoxes  towards  the  east.  This  perturbation  is  exceed- 
ingly minute,  being  only  about  yth  of  a  second  of  arc  in  a 
year.  The  value  50".26  is  the  algebraic  sum  of  all  these 
perturbations. 

119.  Results  of  Precession. — One  result  of  precession   is 
to  make  the  interval  of  time  between  two  successive  returns 
of  the  sun  to  the  vernal  equinox  less  than  the  time  of  one 
sidereal  revolution,  by  the  time  required  by  the  sun  to  pass 
over    50".26,   which  is  20m.   23.6s.     Hence  we   have  the 
tropical  year,  to  which  reference  has  already  been  made  in 
Art.  111.     Another  result  is  that  the  signs  of  the  Zodiac 
(Art.  91)  no  longer  coincide  with  the  constellations  after 
which  they  are  named,  but  have  retreated  towards  the  west 
by  about  28°,  or  nearly  one  sign:  so  that  the  constellation  of 
Aries  now  lies  in  the  sign  of  Taurus.     Still  another  result  is 
that  the  same  star  is  not  the  pole-star  in  different  ages.     Re- 
ferring to  Fig.  46,  the  pole  of  the  heavens,  P,  will  have  re- 
volved about  A  to  the  position  G,  in  the  course  of  about 
13,000  years;  and  a  star  of  the  first  magnitude,  called  Vega, 
which  is  now  about  51°  from  the  pole,  will  at  that  time  be 
less  than  5°  from  the  pole,  and  will  be  the  pole-star. 

120.  Nutation. — Since  precession  is  the  result  of  the  ten- 
dency of  the  sun  to  change  the  position  of  the  plane  of  the 
equator,  it  is  evident  that  there  will  be  no  precession  when 
the  sun  is  itself  in  the  plane  of  the  equator, — in  other  words, 
at  the  equinoxes, — and  that  the  precession  will  be  at  its 
maximum  when  the  sun  is  the  farthest  from  the  plane  of  the 
equator:  that  is  to  say,  is  at  the  solstices.     The  amount  of 
precession  due  to  the  influence  of  the  moon  is  subject  to  a 
similar  variation,  being  the  greatest  when  the  moon's  decli- 
nation is  the  greatest.     The  result  is  that  the  pole  of  the 
heavens  has  a  small  oscillatory  motion  about  its  mean  place. 
This  motion  is  called  nutation.     If  the  effect  of  nutation 
could  be  separated  from  that  of  precession,  the  pole  would  be 
found  to  move  in  a  very  minute  ellipse,  having  a  major  axis 
of  18".5,  and  a  minor  axis  of  13". 7,  the  period  of  one  revolu- 


124  LAW  OF  UNIVERSAL  GRAVITATION 

tion  in  this  ellipse  being  very  nearly  18.6  years.     Since, 
however,  these  two  perturbations  co-exist,  the  result  is  that 
the  pole  of  the  heavens  revolves  about 
the  pole  of  the  ecliptic  in  an  undulating 
curve,  as  shown  in  Fig.  48.     Practically, 
nutation  depends  upon  the  moon  only; 
the  period  of  18.6  years  being  that  of  the 
revolution  of  the  moon's  nodes  (Art.  128). 
121.  Change  in  the  Obliquity  of  the 
FIG.  48.  Ecliptic. — It  was  stated  in  Art.  118  that 

the  latitudes  of  the  stars  were  found 
to  vary  from  year  to  year  by  a  very  minute  amount.  This 
change  in  the  latitudes  is  due  to  a  change  in  the  position  of 
the  plane  of  the  ecliptic,  involving  a  change  in  the  obliquity 
of  the  ecliptic.  The  obliquity  of  the  ecliptic  in  1919  was 
23°  26'  59":  and  the  annual  amount  of  diminution  to  which 
it  is  now  subject  is  0".47.  Mathematical  investigations 
show  that  after  certain  moderate  limits  have  been  reached 
this  diminution  will  cease,  and  the  obliquity  will  begin  to 
increase.  The  arc  through  which  the  obliquity  oscillates  is 
about  1°  21',  and  the  time  of  one  oscillation  is  about  ten 
thousand  years. 

122.  Advance  of  the  Line  of  Apsides. — The  line  connect- 
ing the  earth's  perihelion  and  aphelion  is  called  the  line  oj 
apsides.     This  line  revolves  from  west  to  east,  with  an  annual 
rate  of  11". 8  a  perturbation  due  to  the  attraction  exerted 
on  the  earth  by  the  superior  planets.     The  time  in  which  the 
earth   moves  from   perihelion   to   perihelion   is   called   the 
anomalistic  year  (from  anomaly,  Art.  98).     It  is  evidently 
longer  than  the  sidereal  year,  and  is  found  to  contain  365d. 
6h.   13m.  53.1s. 

ABERRATION 

123.  The  apparent  direction  of  a  celestial  body  is  deter- 
mined by  the  direction  of  the  telescope  through  which  it  is 
observed.     In  consequence  of  the  motion  of  the  earth,  and 


ABERRATION  125 

the  progressive  motion  of  light,  the  telescope  is  carried  to  a 
new  position  while  the  light  is  descending  through  it,  and 
therefore  the  apparent  direction  of  the  body  will  differ  from 
its  true  direction. 

In  Fig.  49,  let  OF  be  the  position  of  the  axis  of  a  telescope 
at  the  instant  when  the  rays  of  light  from  the  star  S  reach 
the  object  glass  0.  The  rays,  after  passing  through  the  glass, 
begin  to  converge  towards  a  fixed  point  in  space,  with  which, 
at  this  instant,  the  intersection  of  the  cross-wires  coincides. 
Let  the  earth  be  moving  in  the 
direction  FA.  Since  the  trans- 
mission  of  light  is  not  instan- 
taneous,  time  is  required  for  the 
light  to  pass  from  0  to  the  fixed 
point  in  space,  and  in  that  time 
the  earth  will  carry  the  axis  of 
the  telescope  to  some  new  posi- 
tion O'F'.  The  cross-wires  will 
then  be  at  Fr,  while  the  rays, 
whose  motion  in  space  is  entirely  FIG.  49. 

independent   of    any    motion   of 

the  telescope,  will  tend  to  meet  at  the  point  F.  In  order, 
then,  to  have  the  image  of  the  star  coincide  with  the 
intersection  of  the  wires,  the  telescope  must  be  so  moved 
that  its  axis  will  lie  in  the  position  O'F.  The  star  will 
then  appear  to  lie  in  the  direction  FS',  while  its  true 
direction  is  of  course  FS:  and  the  angle  which  these  two 
directions  make  with  each  other,  or  the  angle  F'O'F,  is 
called  the  aberration.  Representing  this  angle  by  A,  and 
the  angle  O'FF',  the  angle  between  the  apparent  direction 
of  the  star  and  the  direction  in  which  the  earth  is  moving, 
by  7,  we  shall  have, 

sin  A:  sin  I  =  FF'  :  O'F'. 

But  the  ratio  FF'\  O'F'  is  the  ratio  between  the  velocity  of 
the  earth  and  that  of  light :  so  that  the  sine  of  the  aberration  is 


126  LAW  OF  UNIVERSAL  GRAVITATION 

equal  to  the  ratio  of  the  velocity  of  the  earth  to  that  of  light, 
multiplied  by  the  sine  of  the  angle  /. 

124.  Diurnal  Aberration. — Aberration  causes  the  celestial 
bodies  to  appear  to  be  nearer  than  they  really  are  to  that 
point  of  the  celestial  sphere  towards  which  the  motion  of  the 
earth  is  directed  at  the  instant  of  observation.     As  a  correc- 
tion, then,  it  is  to  be  applied  in  the  opposite  direction.     There 
is  evidently  no  aberration  when  the  motion  of  the  earth  is 
directly  towards  the  star,  and  the  greatest  amount  of  aberra- 
tion occurs  when  the  direction  of  the  earth's  motion  is  at 
right  angles  to  the  direction  of  the  star. 

Aberration  is  of  two  kinds,  corresponding  to  the  daily 
and  the  yearly  motion  of  the  earth.  The  diurnal  aberration 
tends  to  displace  all  bodies  in  the  direction  in  which  the 
earth  is  carrying  the  observer:  that  is  to  say,  in  an  easterly 
direction.  It  evidently  varies  with  the  linear  velocity  of  the 
observer,  and  is  therefore  the  greatest  at  the  equator  and 
zero  at  the  poles.  Owing  to  the  minuteness  of  the  velocity 
of  any  point  of  the  earth's  surface  about  the  axis  in  compar- 
ison with  the  velocity  of  light,  the  diurnal  aberration  is 
extremely  small,  its  greatest  value  being  less  than  J  of  a 
second  of  arc. 

125.  Annual  Aberration. — The  displacement  of  a  star 
occasioned  by  the  motion  of  the  earth  in  its  orbit  about  the 
sun  is  called  the  annual  aberration.     The  effect  which  it  has 
on  the  motion  of  any  body  will  depend  on  the  relative  situa- 
tion of  that  body  to  the  plane  of  the  ecliptic,  as  may  be  seen 
in  Fig.  50.     In  this  figure  S  represents  the  sun,  A  BCD  the 
orbit  of  the  earth,  and  K  the  pole  of  the  ecliptic.     Suppose 
a  star  to  be  at  K.     As  the  earth  moves  through  A,  in  the 
direction  indicated  by  the  arrow,  the  star  will  be  displaced 
from  K  to  a;  as  the  earth  moves  through  J5,  the  star  will  be 
seen  at  6,  etc.     Since  the  direction  of  this  star  is  always  at 
right  angles  to  the  direction  in  which  the  earth  is  moving, 
the  aberration  will  continually  be  at  its  maximum,  as  shown 
in  the  previous  article,  and  the  star  will  describe  a  circle  about 


ABERRATIONS 


127 


d' 


its  true  place  as  a  center.  If  the  star  is  in  the  plane  of  the 
ecliptic,  as  at  s,  there  will  be  no  aberration  when  the  earth 
is  at  A  or  C,  and  the 
aberration  will  be  at 
its  maximum  when  the 
earth  is  at  B  or  D. 
The  star  will  therefore 
during  the  year  de- 
scribe the  arc  b'd', 
equal  in  value  to  twice 
the  maximum  of  aber- 
ration, and  having  the 
true  place  of  the  star  B 
at  its  middle  point. 
If  the  star  is  situated  < 
between  the  pole  and 
the  plane  of  the  eclip- 
tic, it  will  describe  an 

ellipse,  the  semi-major  axis  of  which  is  the  maximum  of 
aberration,  and  the  semi-minor  axis  of  which  increases  with 
the  latitude  of  the  star. 

126.  Velocity  of  Light. — The  maximum  value  of  aberra- 
tion is  the  same  for  all  bodies,  and  may  be  obtained  by  ob- 
serving the  apparent  motion  of  a  fixed  star  during  the  year. 
Its  value  has  thus  been  obtained,  and  is  20". 47.  Now,  since 
the  maximum  of  aberration  occurs  when  the  angle  /,  in  the 
formula  in  Art.  123,  is  90°,  we  shall  have,  denoting  this  maxi- 
mum by  A' 

sin  A'  = 


c 
FIG  50. 


O'F'' 


But  FFf  is  the  velocity  of  the  earth  in  its  orbit,  or  18.4  miles 
a  second.  Hence,  the  velocity  of  light  in  a  second  is  18.4 
miles  multiplied  by  the  cosecant  of  20".47,  which  will  be 
found  to  be  186,324  miles.  Experiments  of  "a  totally  dif- 
ferent character  have  given  almost  precisely  the  same  result; 


128  LAW  OF  UNIVERSAL  GRAVITATION 

and  it  is  believed  that  this  estimate  is  within  a  thousand 
miles  of  the  true  velocity. 

If  we  divide  the  distance  of  the  earth  from  the  sun  by  this 
velocity,  we  find  that  it  requires  8m.  18.58s.  for  light  to  pass 
over  that  distance.  When  we  look  at  the  sun,  therefore,  we 
see  it,  not  as  it  is  at  the  time  of  observation,  but  as  it  was 
8m.  18.58s.  previously;  and  in  the  same  way  every  other 
celestial  body  appears  to  be  in  a  different  position  from  that 
which  it  really  occupies  at  the  instant  we  observe  it.  The 
apparent  place  of  a  planet  is  affected  also  by  this  considera- 
tion: that  the  planet,  moving  in  its  orbit  about  the  sun, 
changes  its  position  in  its  orbit  during  the  interval  of  time  in 
which  light  passes  from  the  planet  to  the  earth.  Hence  we 
have  what  is  called  planetary  aberration.*  It  may  be  well  to 
notice  that  aberration  proper  and  planetary  aberration  are 
taken  into  account  in  computing  the  apparent  places  of  the 
celestial  bodies  as  given  in  the  Nautical  Almanac.  The 
apparent  longitude  of  the  sun,  for  instance,  is  20". 47  less  than 
its  true  longitude;  and  similar  corrections  are  made  in  deter- 
mining its  apparent  right  ascension  and  declination. 

127.  Aberration  a  Proof  of  the  Earth's  Revolution  about 
the  Sun. — The  existence  of  the  phenomenon  of  aberration,  as 
described  in  Art.  125,  is  a  matter  of  undoubted  observation : 
and  when  the  close  agreement  of  the  velocity  of  light  obtained 
in  the  preceding  article  with  the  velocity  obtained  by  indepen- 
dent philosophical  experiments  is  taken  into  consideration,  it 
is  fair  to  regard  the  existence  of  aberration  as  a  strong  direct 
proof  of  the  revolution  of  the  earth  about  the  sun.  Another 
proof,  similar  in  many  respects  to  this,  will  be  noticed  when 
we  come  to  the  subject  of  the  eclipses  of  Jupiter's  satellites. 

*  Herschel  suggests  (Outlines  of  Astronomy,  §  335)  that  this  might  be 
called  the  equation  of  light,  in  order  to  prevent  its  being  confounded  with 
the  real  aberration  of  light. 


CHAPTER  IX 
THE  MOON 

128.  The  Orbit  of  the  Moon. — While  the  moon,  in 
common  with  all  the  celestial  bodies,  has  the  apparent  west- 
ward motion  which  is  due  to  the  rotation  of  the  earth,  it  also 
changes  its  relative  position  to  the  other  bodies,  and  is  con- 
tinually falling  behind,  or  to  the  east  of  them,  in  this  diurnal 
motion.  In  other  words,  it  has  an  independent  motion, 
either  real  or  apparent,  from  west  to  east.  This  eastward 
motion  is  so  rapid  that  we  only  need  to  observe  the  relative 
situations  of  the  moon  and  some  conspicuous  star,  during 
a  few  hours  on  any  favorable  night,  to  notice  a  perceptible 
change  in  their  angular  distance.  If  the  right  ascension  and 
the  declination  of  the  moon  are  determined  from  day  to  day, 
precisely  as  the  same  elements  of  the  sun's  position  were 
determined  (Art.  89),  and  the  corresponding  positions  are 
laid  down  upon  a  celestial  globe,  we  shall  find  that  the  moon 
makes  a  complete  revolution  in  the  heavens,  about  the  earth 
as  a  center,  in  an  average  period  of  27d.  7h.  43m.  11.5s.  We 
shall  also  find  that  the  plane  of  the  moon's  orbit  intersects 
the  plane  of  the  ecliptic  at  an  angle  whose  mean  value  is 
5°  8'  40",  and  in  a  line  which,  like  the  earth's  line  of  equi- 
noxes, is  continually  revolving  towards  the  west:  so  that  the 
apparent  orbit  of  the  moon  is  not  a  circle,  but  a  kind  of  spiral. 
This  revolution  is  much  more  rapid,  however,  in  the  case  of 
the  moon,  the  amount  of  retrogradation  being  about  1°  27' 
in  a  month,  and  the  complete  revolution  being  effected  in 
18.6  years. 

The  movement  of  the  moon  in  its  orbit  is  represented  in 

129 


130  THE  MOON 

Fig.  51.  Let  E  be  the  earth,  and  the  circle  MANC  the 
plane  of  the  ecliptic.  Let  the  moon  be  at  M  at  any  time. 
Then  will  MN  be  the  line  in  which  the  plane  of  the  moon's 
orbit  intersects  the  plane  of  the  ecliptic.  This  line  is  called 
the  line  of  the  nodes.  That  extremity  of  the  line  through 
which  the  moon  passes  in  moving  from  the  southern  to  the 

northern  side  of  the  ecliptic  is 
called  the  ascending  node,  the 
other  the  descending  node.  Let 
the  moon  move  on  from  M  in 
the  arc  MB.  When  it  descends 
to  the  ecliptic,  it  will  meet  it, 
not  at  the  point  N,  but  at  some 
point  TV',  and  the  line  of  the 
nodes  will  take  the  new  posi- 
tion N'M' '.  The  moon  moves 
on  to  the  other  side  of  the 

ecliptic,  passes  through  the  arc  N'Dt  and  when  it  again 
returns  to  the  ecliptic,  will  meet  it,  not  at  M',  but  at  some 
point  M",  and  the  line  of  the  nodes  will  take  the  position 
M"N" .  The  revolution  of  the  line  of  the  nodes  is  evidently 
in  an  opposite  direction  to  that  in  which  the  moon  itself 
revolves,  and  is  therefore  from  east  to  west. 

129.  Cause  of  the  Retrogradation  of  the  Nodes.— This 
retrograde  movement  of  the  moon's  nodes  is  similar  in 
character  to  the  precession  of  the  equinoxes,  and  is  due  to  the 
attraction  which  the  sun  exerts  upon  the  moon.  Since  the 
plane  of  the  moon's  orbit  is  inclined  to  the  plane  of  the 
ecliptic,  the  attraction  of  the  sun  will,  in  general,  tend  to 
draw  the  moon  out  of  its  orbit  towards  the  ecliptic.  The 
only  exceptions  to  this  rule  will  occur  when  the  moon  is  at 
one  of  its  nodes,  and  is  therefore  in  the  plane  of  the  ecliptic, 
and  also  when  the  line  of  the  moon's  nodes  passes  through 
the  sun,  at  which  time  the  attraction  of  the  sun  is  exerted 
along  this  line,  and  consequently  in  the  plane  of  the  moon's 
orbit.  In  Fig.  51  let  the  moon  be  at  B,  and  the  sun  anywhere 


CHANGE  IN  OBLIQUITY  OF  MOON  S  ORBIT         131 

in  the  ecliptic  except  on  the  line  of  the  nodes.  As  the  moon 
moves  on,  the  sun  is  continually  drawing  it  down  to  the 
ecliptic,  and  it  will  hence  meet  the  ecliptic,  not  at  N,  but  at 
N'.  The  same  effect  will  be  seen  at  every  other  position  of 
the  moon  in  its  orbit,  with  only  the  exceptions  already 
mentioned. 

130.  Change  in  the  Obliquity  of  the  Moon's  Orbit. — 
It  is  evident  from  the  same  figure  that  the  angle  which  the 
arc  BN'  makes  with  the  plane  of  the  ecliptic  is  greater  than 
the  angle  which  an  arc  drawn  through  B  and  N  would  make. 
The  obliquity  of  the  plane  of  the  moon's  orbit  is  therefore 
increased  as  the  moon  approaches  the  node.     It  may  be 
shown  in  the  same  way  that  the  obliquity  is  diminished  as 
the  moon  recedes  from  the  node.     The  extreme  limits  which 
this  angle  attains  are  5°  20'  6"  and  4°  57'  22". 

131.  Elliptical  Form  of  the  Moon's  Orbit.— The  angular 
diameter  of  the  moon  varies  at  different  points  of  its  orbit, 
while  its  mean  value  remains  the  same  from  month  to  month; 
we  therefore  conclude,  as  we  concluded  in  the  case  of  the  sun, 
that  its  distance  from  the  earth  is  not  constant,  the  greatest 
distance  corresponding  to  the  least  diameter,   and  the  least 
distance  to  the  greatest  diameter.     If  we  neglect  the  retro- 
gradation  of  the  moon's  nodes,  and  represent  graphically  the 
moon's  orbit  by  a  method  identical  with  the  method  em- 
ployed in  representing  the  earth's  orbit  (Art.  98),  we  shall 
find  the  orbit  to  be  an  ellipse,  with  the  earth  at  one  of  the 
foci.     The  eccentricity  of  the  ellipse  is  0.0549,  or  very  nearly 
Ath. 

132.  Line  of  Apsides. — That  point  in  the  moon's  orbit 
where  it  is  the  nearest  to  the  earth  is  called  the  perigee,  and 
that  point  where  it  is  the  farthest  from  the  earth,  the  apogee. 
The  line  connecting  these  two  points  is  called  the  line  of 
apsides.     It  is  also  the  major  axis  of  the  moon's  orbit.     This 
line  revolves  in  the  plane  of  the  moon's  orbit  from  west  to 
east,  making  a  complete  revolution  in  very  nearly  nine  years. 

The  following  description  of  the  moon's  orbit,  and  of  the 


132  THE  MOON 

changes  to  which  it  is  subject,  is  given  by  Herschel  in  his 
Outlines  of  Astronomy.  "The  best  way  to  form  a  distinct 
conception  of  the  moon's  motion  is  to  regard  it  as  describing 
an  ellipse  about  the  earth  in  the  focus,  and  at  the  same  time 
to  regard  this  ellipse  itself  to  be  in  a  twofold  state  of  revolu- 
tion; first,  in  its  own  plane,  by  a  continual  advance  of  its  own 
axis  in  that  plane;  and  second,  by  a  continual  tilting  motion 
of  the  plane  itself,  exactly  similar  to,  but  much  more  rapid 
than  that  of  the  earth's  equator." 

133.  Variation  in  the  Moon's  Meridian  Zenith  Distance. 
—  From  the  formula  in  Art.  76  we  have. 


At  any  place,  then,  the  latitude  remaining  constant,  the  least 
meridian  zenith  distance  will  occur  when  the  moon's  declina- 
tion has  the  same  name  as  the  latitude,  and  is  at  its  maxi- 
mum, and  the  greatest  will  occur  when  the  declination  has 
the  opposite  name,  and  is  also  at  its  maximum.  Since  the 
plane  of  the  moon's  orbit  is  inclined,  at  the  most,  5°  20'  to 
the  plane  of  the  ecliptic  (Art.  130)  ,  the  greatest  value  of  the 
•declination,  either  north  or  south,  is  5°  20'+23°  27'.  The 
variation  in  the  meridian  altitude  will  therefore  be  double 
this  amount,  or  57°  34'.  At  Annapolis,  in  latitude  38°  59'  N., 
the  greatest  altitude  is  79°  48',  the  least  22°  14'.  There  is 
an  exception  to  this  general  rule  in  the  case  of  those  places 
whose  latitude  is  less  than  28°  47'  :  since  at  those  places  the 
greatest  altitude  occurs  when  the  moon  is  in  the  zenith,  or, 
as  is  evident  from  the  formula,  when  the  declination  is  equal 
to  the  latitude,  and  has  the  same  name. 

The  new  moon  is  in  the  same  part  of  the  heavens  that  the 
sun  is  in  (Art.  139)  ,  and  the  full  moon  is  in  the  opposite  part. 
Since  the  sun  attains  its  least  altitude  in  winter  and  its  great- 
est in  summer,  new  moons  will  run  low  in  winter  and  full 
moons  will  run  high:  while  in  summer  the  opposite  of  this 
will  take  place. 


DISTANCE,  SIZE,  AND  MASS  OF  THE  MOON         133 

DISTANCE,  SIZE,  AND  MASS  OF  THE  MOON 

134.  Since  the  sine  of  the  moon's  horizontal  parallax  is 
the  ratio  of  the  radius  of  the  earth  to  the  distance  of  the 
moon  from  the  earth,  it  is  evident  that  we  can  determine  this 
distance  as  soon  as  we  obtain  the  horizontal  parallax.  The 
horizontal  parallax  may  be  found  in  the  following  manner: 
In  Fig.  52,  let  0  be  the  center  of  the  earth,  EQ  its  equator, 
and  A  and  B  the  positions  of  two  observers  on  the  same 
meridian,  whose  zeniths  are  Z  and  Zf.  Let  M  be  the  moon's 
position  when  crossing  the  meridian.  The  apparent  zenith 


distance  at  A,  corrected  for  refraction,  is  the  angle  ZAM, 
the  geocentric  zenith  distance  is  ZOM,  and  the  difference  of 
these  two  angles,  the  angle  A  MO,  is  the  parallax  in  altitude. 
In  the  same  way  OMB  is  the  parallax  at  B.  Represent  the 
parallax  at  A  by  p,  that  at  B  by  pf,  the  horizontal  parallax 
by  P,  the  apparent  meridian  zenith  distance  at  A  by  2,  and 
that  at  B  by  z'. 

We  have,  by  Geometry, 

p  =  z-AOM, 

p'=z'-BOM, 

and,  consequently, 

'  =  z+z'-AOB. 


134  THE  MOON 

We  have  also,  from  Art.  54, 

p=P  sin  z, 

?/  =  Psinz', 
and.  therefore, 

p-\-pf  =  P  (sin  z+sin  z')  . 

Combining  the  two  equations,  and  finding  the  expression 
for  P, 


sin  z+sin  z'* 

But  AOB  is  evidently  the  difference  of  latitude  of  A  and  B. 
We  have,  then,  as  our  method  of  finding  the  moon's  hori- 
zontal parallax,  to  subtract  the  difference  of  latitude  of  the 
two  places  from  the  sum  of  the  apparent  zenith  distances, 
and  to  divide  the  remainder  by  the  sum  of  the  sines  of  the 
two  zenith  distances. 

It  is  important  that  the  two  places  of  observation  shall 
differ  widely  in  latitude.  It  is  not,  however,  necessary 
that  they  shall  be  on  the  same  meridian,  since,  either  from 
tables  of  the  moon's  motion,  or  from  actual  observation  on 
successive  days  before  and  after  the  time  of  observation,  we 
can  obtain  the  change  of  meridian  zenith  distance  correspond- 
ing to  any  known  difference  of  longitude,  and  thus  reduce  the 
two  observed  zenith  distances  to  the  same  meridian. 

135.  The  Moon's  Horizontal  Parallax.  —  By  observations 
similar  to  those  above  described,  the  mean  value  of  the 
moon's  equatorial  horizontal  parallax  is  found  to  be  57'  2".  6. 
The  mean  distance  of  the  moon  from  the  earth  is  therefore 
3963.34  miles  multiplied  by  the  cosecant  of  57'  2."6,  or 
238,862  miles.  The  horizontal  parallax  varies  between  the 
limits  of  61'  27"  and  53'  55",  and  the  distance  between 
257,900  and  221,400  miles.  It  must  be  noticed  that  by  the 
mean  value  of  the  horizontal  parallax  given  above  is  not 
meant  the  half  sum  of  the  two  extreme  values,  but  the  value 
which  the  parallax  has  when  the  moon  is  at  its  mean  distance 
from  the  earth. 


DISTANCE,  SIZE,  AND  MASS  OF  THE  MOON        135 

136.  Size    of     the    Moon.  —  The    angular    semi-diam- 
eter of  the  moon  at  its  mean  distance  from   the   earth   is 
found  to  be  15'  32//6.     Its  linear  semi-diameter  is  therefore 
obtained  by  multiplying  the  mean  distance  by  the  sine  of 
15'  32."6,  and  is  found  to  be  1080  miles,  or  about  T3Tths  of 
the  radius  of  the  earth.     The  volumes  of  two  spheres  being 
to  each  other  as  the  cubes  of  their  radii,  the  volume  of  the 
moon  will  be  found  to  be  about  ^th  of  that  of  the  earth. 

137.  Mass  of  the  Moon. — "The  mass  of  the  moon  is 
concluded,  first,  from  the  proportion  of  the  lunar  to  the  solar 
tide,  as  observed  at  various  stations,  the  effects  being  sepa- 
rated from  each  other  by  a  long  series  of  observations  of  the 
relative  heights  of  spring  and  neap  tides,  which,  we  have 
seen,  depend  on  the  proportional  influence  of  the  two  lu- 
minaries;   secondly,    from    the    phenomenon    of    nutation, 
which,  being  the  result  of  the  moon's  attraction  alone,  affords 
a  means  of  calculating  its  mass,  independent  of  any  knowl- 
edge of  the  sun's"  (Herschel,  Outlines  of   Astronomy).     A 
discussion  of  the  cause  of  the  tides  will  be  found  in  Chapter 
XI.    With  regard  to  the  second  method,  we  have  already 
seen  (Art.  120)  that  nutation  depends  theoretically  on  both 
the  moon  and  the  sun ;  practically,  on  the  moon  only.     Other 
methods  of  determining  the  moon's  mass  might  be  given,  all 
depending  on  its  attractive  power.     The  mass  found  by  these 
different  methods  is  about  -g^st  of  that  of  the  earth. 

The  density  of  the  moon,  being  directly  as  the  mass,  and 
inversely  as  the  volume,  will  be  ff,  °r  about  fths  of  the 
density  of  the  earth. 

138.  Augmentation   of   the   Moon's    Semi-Diameter. — 
If  at  any  time  we  measure  the  angular  semi-diameter  of  the 
moon,  we  shall  find  that  it  increases  with  the  moon's  altitude, 
being  least  when  the  moon  is  in  the  horizon  and  greatest 
when  in  the  zenith.     This  increase  is  explained  in  Fig.  53. 
Let  E  be  the  center  of  the  earth,  and  M  that  of  the  moon. 
With  the  distance  between  E  and  M  as  a  radius,  describe  the 
semi-circumference    AM'B.     When    the    moon    is    in    the 


136  THE  MOON 

horizon  of  the  point  C,  its  distances  from  C  and  from  E  are 
very  nearly  equal.  But  as  the  moon  rises,  the  distance 
CM  continually  decreases,  while  EM ,  the  distance  of  the 
moon  from  the  earth's  center,  remains  sensibly  constant. 

When  the  moon  is  in 
'  the    zenith,  or  at  Mf, 

the  distance  CM'  is  less 
than  EM'  by  the  radius 
of  the  earth.  Now,  the 
angular  semi-diameter 
of  the  moon  will  in- 
crease, as  shown  in  Art. 
98,  very  nearly  as  the 
FIG.  53.  distance  of  the  moon 

from  the  observer  de- 
creases. But  the  earth's  radius  is  about  ^Vth  of  the 
distance  of  the  moon  from  the  earth's  center:  therefore 
the  semi-diameter  of  the  moon  in  the  zenith  will  be  greater 
than  the  semi-diameter  in  the  horizon  by  j^ih  of  itself,  or  by 
about  15".  This  increase  is  called  the  augmentation  of  the 
moon's  semi-diameter. 

THE  MOON'S  PHASES 

139.  Two  bodies  are  said  to  be  in  conjunction  when  they 
have  the  same  longitude.  They  are  said  to  be  in  opposition 
when  their  longitudes  differ  by  180°;  and  in  quadrature  when 
their  longitudes  differ  by  either  90°  or  270°. 

The  moon  is  an  opaque  body,  which  is  rendered  visible  to 
us  by  the  rays  of  light  which  it  reflects  from  the  sun.  The 
phases  of  the  moon  are  due  to  the  different  relative  positions 
to  the  sun  and  the  earth  which  it  has  while  revolving  about 
the  earth. 

In  Fig.  54  let  E  be  the  earth,  and  the  circle  ACFH  the 
orbit  of  the  moon.  Since  the  inclination  of  the  plane  of  the 
moon's  orbit  to  the  plane  of  the  ecliptic  is  only  a  few  degrees, 


THE  MOON'S  PHASES 


137 


we  may  neglect  it  in  this  case,  and  suppose  the  two  planes  to 
coincide.  Let  the  sun  lie  in  the  direction  ES.  Since  the 
distance  of  the  sun  from  the  earth  is  about  387  times  the 
distance  of  the  moon  from  the  earth,  the  lines  ES,  HS,  BS, 
etc.,  drawn  to  the  sun  from  different  points  of  the  moon's 
orbit,  may  be  considered  to  be  sensibly  parallel.  Let  us 
first  suppose  the  moon  to  be  in  conjunction  with  the  sun  at 
the  point  A.-  Here  only  the  dark  portion  of  the  moon  is 
turned  towards  the  earth,  and  the  moon  is  therefore  invisible. 


FIG.  54. 

This  is  called  new  moon.  As  the  moon  moves  on  towards 
B,  the  enlightened  part  begins  to  be  visible,  and  when  it 
reaches  C,  90°  in  longitude  from  the  sun,  half  the  enlightened 
part  is  visible,  and  the  moon  is  at  its  first  quarter.  When 
the  moon  is  at  F,  in  opposition  to  the  sun,  all  the  illuminated 
part  is  turned  towards  the  earth,  and  the  moon  is  full.  The 
moon  wanes  after  leaving  F,  passes  through  its  last  quarter 
at  H,  and  finally  becomes  again  invisible. 

Between  A  and  C  the  moon  is  crescent,  as  represented  at 
L,  and  between  C  and  F  it  is  gibbous,  as  represented  at  N. 


138 


THE  MOON 


The  same  terms  are  also  applied  to  the  appearance  of  the 
moon  between  H  and  A  and  between  F  and  H. 

140.  Phases  of  the  Earth  to  the  Moon. — It  is  evident  from 
Fig.  54  that  the  earth  presents  phases  to  the  moon  identical 
in  character  with  those  presented  by  the  moon  to  the  earth, 
although  similar  phases  are  not  presented  by  each  body  at 
the  same  time.  Thus  at  the  time  of  new  moon  the  earth  is 
full  to  the  moon:  and  the  light  which  it  then  reflects  to  the 
moon  renders  the  unenlightened  part  of  the  moon  faintly 
visible  to  the  earth.  As  the  moon  moves  on  to  its  first 
quarter,  the  earth  reflects  less  and  less  light  to  it,  until 
finally  the  unenlightened  portion  disappears. 


SIDEREAL  AND  SYNODICAL  PERIODS 

141.  The  sidereal  period  of  the  moon  is  the  interval  of  time 
in  which  it  makes  one  complete  revolution  in  its  orbit  about 

the  earth.  The  synodi- 
cal  period  (or  lunation) 
is.  the  interval  between 
two  successive  conjunc- 
tions or  oppositions. 
Owing  to  the  earth's 
revolving  about  the  sun, 
and  carrying  the  moon 
with  it,  the  synodical 
period  is  longer  than  the 
sidereal  period,  as  may 
be  seen  in  Fig.  55. 

Let  S  be  the  sun,  E 
FIG.  55.  the  earth,  and  MANE 

the   orbit  of  the  moon. 

Let  the  moon  be  at  M,  in  conjunction  with  the  sun.  As 
the  moon  moves  about  E  in  the  curve  MANB,  the 
earth  also  moves  about  the  sun  in  the  direction  EEf.  The 
next  conjunction  will  therefore  not  occur  until  the  moon 


SIDERIAL  AND  SYNODICAL  PERIODS  139 

reaches  Mff.  Now,  if  through  E'  we  draw  the  line  M'N' 
parallel  to  MN,  the  sidereal  period  of  the  moon  is  completed 
when  the  moon  reaches  M' '.  The  synodical  period  is  there- 
fore greater  than  the  sidereal  period  by  the  time  required 
by  the  moon  to  pass  through  the  angle  M'E'M".  This  angle 
is  evidently  equal  to  the  angle  ESE',  which  is  the  angular 
advance  of  the  earth  in  its  orbit  in  the  period  of  one  synodical 
revolution  of  the  moon.  In  one  lunar  month,  then,  the 
angular  advance  of  the  moon  in  its  orbit  is  greater  by  360° 
than  the  angular  advance  of  the  earth  in  its  orbit.  If,  there- 
fore, we  denote  the  moon's  sidereal  period  in  days  by  P,  its 
synodical  period  by  S,  and  the  earth's  sidereal  period,  or  one 
sidereal  year,  by  L,  we  shall  have, 

360° 
~   =the  earth's  daily  angular  velocity; 

360° 

=  the  moon's  daily  angular  velocity; 


P 
360° 


=  the  moon's  daily  angular  gain  on  the  earth. 


S 

Hence  we  shall  have, 

360°     360° _ 360° 
P         T      ~S~' 

ST 


:.  P= 


s+r 


The  sidereal  period  of  the  moon  is  therefore  obtained  by 
multiplying  the  sidereal  year  by  the  moon's  synodical  period, 
and  dividing  the  product  by  the  sum  of  the  sidereal  year,  and 
the  synodical  period. 

142.  Values  of  the  Synodical  and  Sidereal  Periods. — 
The  value  of  the  synodical  period  is  not  constant,  but  varies 
from  month  to  month.  A  mean  value  may,  however,  be 
obtained  by  dividing  the  interval  of  time  between  two  oppo- 


140  THE  MOON 

sitions,  not  consecutive,  by  the  number  of  sy nodical  revo- 
lutions in  that  interval.  Now,  the  day,  the  hour,  and  even 
the  probable  minute,  at  which  an  opposition  to  the  moon 
occurred  in  the  year  720  B.C.,  were  recorded  by  the  Chal- 
daeans;  and  by  comparing  this  time  with  the  results  of  recent 
observations,  an  extremely  accurate  value  of  the  mean 
synodical  period  is  obtained.  It  is  found  to  be  29d.  12h. 
44m.  02.8s.  We  have,  then,  for  the  value  of  the  sidereal 
period,  by  the  formula  in  the  preceding  article, 

365.256  X  29.53  , 
"365.256+29.53      ys: 

whence  we  obtain  the  value  already  given  in  Art.  128. 

143.  Retardation  of  the  Moon,  and  the  Harvest  Moon. 

— The  mean  daily  motion  of  the  moon  towards  the  east  is 
about  13°,  while  that  of  the  sun  is,  as  we  have  already  seen, 
about  1°;  hence  the  moon  is  continually  falling  to  the  rear 
of  the  sun  in  apparent  westward  motion,  and  the  interval  of 
time  between  any  two  successive  transits  of  the  moon  is 
greater  than  the  similar  interval  in  the  case  of  the  sun.  The 
moon,  therefore,  rises  later  and  sets  later,  day  by  day.  This 
is  called  the  retardation  of  the  moon.  Its  amount  varies 
considerably  in  value,  but  is  on  the  average  about  fifty 
minutes. 

The  less  the  angle  which  the  plane  of  the  moon's  orbit 
makes  with  the  plane  of  the  horizon,  the  less  does  the  advance 
of  the  moon  carry  it  with  reference  to  the  horizon,  and,  con- 
sequently, the  less  is  the  retardation  of  the  moon  in  rising. 
Now,  since  the  moon's  orbit  very  nearly  coincides  with  the 
ecliptic,  the  retardation  in  rising  will  in  general  be  the  least, 
when  the  ecliptic  makes  the  least  angle  with  the  horizon. 
By  reference  to  a  celestial  globe,  it  will  be  seen  that  the 
ecliptic  makes  the  least  angle  with  the  horizon  when  the 
vernal  equinox  is  in  the  eastern  horizon  The  least  retarda- 
tion in  rising,  therefore,  occurs  in  each  month  when  the  moon 


ROTATION,  ITERATIONS,  AND  OTHER  PERTURBATIONS  141 

is  near  the  sign  of  Aries.  This  least  retardation  is  especially 
noticeable  when  it  occurs  at  the  time  of  full  moon.  Now, 
when  the  moon  is  in  Aries,  and  full,  the  sun  must  be  in  Libra, 
or  near  the  autumnal  equinox.  This  occurs  about  the  21st 
of  September.  About  the  time,  then,  of  the  full  moon 
which  occurs  near  the  21st  of  September,  the  moon  will  rise, 
for  two  or  three  nights,  only  about  half  an  hour  later  each 
night.  Usually  this  small  retardation  is  noticed  at  the  times 
of  two  full  moons,  one  in  September  and  the  other  in  October. 
The  first  is  called  the  Harvest  Moon,  the  second  the  Hunter's 
Moon.  All  this  relates  to  the  Northern  Hemisphere. 


ROTATION,  LIBRATIONS,  AND  OTHER  PERTURBATIONS 

144.  Rotation  of  the  Moon. — By  observation  of  the  spots 
upon  the  disc  of  the  moon,  it  is  found  that  very  nearly  the 
same  surface  of  the  moon  is  turned  continually  towards  the 
earth.  The  conclusion  drawn  from  this  fact  is  that  the 
moon  rotates  upon  an  axis  in  the  same  time  in  which  it  re- 
volves about  the  earth,  or  in  27.3  days;  The  plane  in  which 
this  rotation  is  performed  makes  an  angle  of  about  1°  32' 
with  the  plane  of  the  ecliptic. 

If  there  are  any  inhabitants  of  the  moon,  their  day  will  be 
equal  in  length  to  about  twenty-nine  of  our  days,  and  their 
night  to  about  twenty-nine  of  our  nights.  Since  the  plane  of 
the  moon's  equator  is  so  nearly  coincident  with  the  plane  of 
the  ecliptic,  there  will  hardly  be  any  sensible  change  of 
seasons :  or  if  there  is,  the  lunar  day  will  be  the  lunar  summer, 
and  the  night  the  winter.  To  the  inhabitants  of  one  hemi- 
sphere the  earth  will  be  perpetually  invisible,  while  to  the 
inhabitants  of  the  other  hemisphere  it  will  present  the  appear- 
ance of  a  body  very  nearly  stationary  in  their  sky,  exhibiting 
phases  similar  to  those  which  we  see  in  the  moon,  with  a 
radius  nearly  four  times  that  of  the  moon,  and  a  surface 
about  thirteen  times  that  of  the  moon. 


142 


THE  MOON 


145.  Librations. — By  libration  is  meant  an  apparent 
oscillatory  movement  of  the  moon,  which  enables  us,  in  the 
course  of  its  revolution,  to  see  something  more  than  an  exact 
hemisphere 

The  libration  in  longitude  is  due  to  the  fact  that  the  moon's 
rotation  on  its  axis  is  perfectly  uniform,  while  its  motion 
about  the  earth  is  not.  Hence  the  line  drawn  from  the 


center  of  the  earth  to  that  of  the  moon  does  not  always  inter- 
sect the  surface  of  the  moon  at  the  same  point,,  and  we  are 
able  at  times  to  look  a  few  degrees,  east  or  west,  beyond  the 
mean  visible  border.  If,  in  Fig.  56,  A  BCD  represents  the 
earth,  E  its  center,  and  R  the  center  of  the  moon,  the  dotted 
lines  at  N  denote  the  limits  between  which,  as  the  moon  re- 
volves about  the  earth,  the  visible  border  may  deviate  from 
its  mean  position. 


ROTATION,  LIBRAT1ONS,  AND  OTHER  PERTURBATIONS   143 

The  libration  in  latitude  is  due  to  the  fact  that  the  axis  of 
the  moon,  remaining  constantly  parallel  to  itself,  is  not 
perpendicular  to  the  plane  of  the  moon's  orbit,  but  is  inclined 
to  it  at  an  angle  of  about  83°  19'.  We  are  therefore  able  at 
certain  times  to  see  about  6°  41'  beyond  the  north  pole  of 
the  moon,  and  at  other  times  the  same  amount  beyond  the 
south  pole.  Thus  in  Fig.  56,  when  the  moon  is  at  M,  we  can 
see  beyond  the  pole  P,  and  when  the  moon  is  at  0,  beyond  the 
pole  p:  since  in  each  case  we  can  see  nearly  that  portion  of 
the  moon  which  lies  between  the  earth  and  the  circle  a&, 
whose  plane  is  perpendicular  to  the  plane  of  the  moon's  orbit. 

The  diurnal  libration  is  due  to  the  difference  between  that 
hemisphere  of  the  moon  which  is  turned  towards  the  center 
of  the  earth  and  that  which  is  turned  towards  any  point  on 
the  surface.  When,  for  instance,  the  moon  is  at  L,  an 
observer  at  C  will  see  the  same  hemisphere  which  is  turned 
towards  the  earth's  center,  while  an  observer  at  G  will  see  a 
different  one.  The  hemisphere  which  is  turned  towards 
any  observer  when  the  moon  is  rising  will  also  be  different 
from  the  one  which  is  turned  towards  him  when  the  moon  is 
setting.  It  is  evident  in  the  figure  that  the  amount  of  this 
libration  varies  with  the  angle  ERG;  that  is  to  say,  with  the 
moon's  parallax. 

Notwithstanding  all  these  librations,  we  are  able  to  see  in 
all  only  about  f^ths  of  the  moon's  surface,  the  remainder 
being  continually  concealed  from  our  view. 

146.  Other  Perturbations. — Besides  these  librations,  and 
the  perturbations  already  mentioned  (Art.  132),  there  are 
other  perturbations  in  the  moon's  longitude  of  which  only  a 
very  brief  notice  can  here  be  given.  The  greatest  of  these 
perturbations  is  called  evedion,  and  was  discovered  by 
Ptolemy  in  the  second  century.  It  arises  from  the  variation 
in  the  eccentricity  of  the  moon's  orbit,  and  from  the  fluctua- 
tions in  the  general  advance  of  the  line  of  the  apsides.  By  it 
the  moon's  mean  longitude  is  alternately  increased  and 
decreased  by  about  1°  20'.  Another  perturbation  in  the 


'144  THE  MOON 

moon's  motion  is  called  variation.  It  depends  solely  on  the 
angular  distance  of  the  moon  from  the  sun,  and  its  maximum 
is  37'.  The  annual  equation  depends  on  the  variable  distance 
of  the  earth  from  the  sun,  and  amounts  to  ll/.  The  secular 
acceleration  is  an  increase  in  the  moon's  motion  which  has 
been  going  on  for  many  centuries,  at  the  rate  of  about  10" 
a  century.  This  perturbation  is  partly  due  to  the  diminution 
of  the  eccentricity  of  the  earth's  orbit;  and  from  what  has 
been  said  on  that  subject  in  Art.  98,  it  is  evident  that  this 
inequality  will  at  some  distant  day  become  a  secular  retarda- 
tion. (See  Note,  page  170.) 

Ah1  of  these  perturbations  are  satisfactorily  explained  by 
the. investigation  of  what  is  known  as  the  problem  of  the  three 
bodies,  in  which  two  bodies  are  supposed  to  revolve  about 
their  common  center  of  gravity,  according  to  the  law  of 
universal  gravitation,  and  the  effects  of  the  attraction  exerted 
by  a  third  body  upon  the  motions  of  these  two  bodies  are 
made  the  object  of  mathematical  examination. 


THE  LUNAR  CYCLE 

147.  If  we  multiply  the  number  of  days,  hours,  etc.,  in  a 
synodical  period  of  the  moon  (Art.  142)  by  235,  the  product 
will  be  6939d.  16h.  27m.  50s,  Now,  in  a  period  of  nineteen 
civil  years  there  are  either  6939  clays,  or  6940  days,  according 
as  there  are  four  or  five  leap  years  in  that  period.  If,  then, 
in  any  year,  new  moon  occurs  on  any  particular  day  of  the 
month,  the  first  of  January,  for  instance,  it  will  occur  again 
on  the  first  of  January  (or  at  all  events  within  a  few  hours  of 
its  end  or  beginning),  after  an  interval  of  nineteen  years; 
and  all  the  new  moons  and  the  other  phases  will  occur  on 
very  nearly  the  same  days  throughout  the  second  period  of 
nineteen  years  on  which  they  occurred  during  the  first 
period.  This  period  is  called  the  Lunar  Cycle.  It  is  also 
called  the  Metonic  Cycle,  having  been  originally  discovered, 


GENERAL  DESCRIPTION  OF  THE  MOON  145 

432  B.C.,  by  Meton,  an  Athenian  mathematician.  The 
present  lunar  cycle  began  in  1919. 

This  cycle  is  used  in  finding  Easter:  Easter  being  the  first 
Sunday  after  the  full  moon  which  occurs  either  upon  or  next 
after  the  21st  day  of  March. 

The  golden  number  of  any  year  is  the  number  which 
marks  the  place  of  that  year  in  the  cycle.  It  may  be  found 
for  any  year  by  adding  1  to  the  number  of  that  year,  and 
dividing  the  sum  by  19;  the  remainder  (or  19,  if  there  is  no 
remainder)  is  the  golden  number. 

Four  lunar  cycles,  or  seventy -six  civil  years,  constitute 
what  is  called  the  Callippic  cycle. 


GENERAL  DESCRIPTION  OF  THE  MOON 

148.  When  viewed  through  powerful  telescopes,  the 
surface  of  the  moon  is  .found  to  be  made  up  of  mountains, 
valleys,  and  plains,  similar  in  general  appearance  to  those 
that  exist  on  the  earth,  As  a  whole,  however,  the  surface 
of  the  moon  is  much  more  uneven  than  that  of  the  earth. 
The  heights  of  over  1000  lunar  mountains  have  been  meas- 
ured, and  some  of  them  have  been  found  to  exceed  20,000 
feet.  Many  of  these  mountains  bear  the  appearance  of 
having  been  at  one  time  volcanoes,  far  surpassing  in  size  and 
activity  those  on  the  earth.  The  common  belief  among 
astronomers  seems  to  be  that  these  lunar  volcanoes  are  now 
extinct.  Messrs.  Beer  and  Madler,  two  Prussian  astron- 
omers who  have  made  the  moon  their  special  study,  have  de- 
tected no  signs  of  activity  in  any  of  the  volcanoes  which  they 
have  examined.  In  October,  1866,  certain  phenomena  were 
noticed  which  seemed  to  show  that  one  at  least  of  these 
volcanoes,  named  Linne,  is  not  extinct,  but  later  observations 
do  not  confirm  this  suspicion. 

There  are  no  signs  of  the  existence  of  water  on  the  moon. 
Certain  large  dark  patches  are  seen,  which  were  formerly 


146  THE  MOON 

considered  to  be  oceans,  gulfs,  etc.,  and  were  so  named,  but 
increased  telescopic  power  shows  that  they  are  dry  plains. 
It  seems  to  be  still  an  open  question  whether  or  not  the 
moon  has  an  atmosphere.  If  there  is  an  atmosphere,  it 
must  be  of  an  extremely  minute  height  and  density;  for  we 
see  no  clouds  and  no  twilight,  and  there  is  nothing  in  the 
phenomena  of  the  occultations  of  stars  by  the  moon  which 
shows  the  existence  of  even  the  rarest  atmosphere. 


CHAPTER  X 

LUNAR  AND  SOLAR  ECLIPSES.    OCCULTATIONS 

149.  Eclipses. — The  obscuration,  either  partial  or  total, 
of  the  light  of  one  celestial  body  by  another  is  in  astronomy 
termed  an  eclipse.     When  the  earth  comes  between  the  sun 
and  the  moon,  the  light  of  the  sun  is  shut  off  from  the  moon, 
and  we  have  a  lunar  eclipse.     A  lunar  eclipse  can  occur  only 
at  the  time  when  the  moon  is  in  opposition  to  the  sun,  that 
is  to  say,  at  the  time  of  full  moon.     When  the  moon  comes 
between  the  earth  and  the  sun,  the  light  of  the  sun  is  shut  off 
from  the  earth,  and  we  have  a  solar  eclipse.     A  solar  eclipse 
can  occur  only  at  the  time  of  new  moon.     An  eclipse  of  a  star 
or  a  planet  by  the  moon  is  called  an  occullation. 

If  the  orbit  of  the  moon  lay  in  the  plane  of  the  ecliptic,  a 
lunar  and  a  solar  eclipse  would  occur  in  every  month. 
Owing,  however,  to  the  inclination  of  the  plane  of  the  moon's 
orbit  to  the  plane  of  the  ecliptic,  the  latitude  of  the  moon  is 
usually  too  great  to  allow  either  kind  of  eclipse  to  take  place; 
and  it  is  only  in  special  cases,  when  the  moon  is  in  or  near  the 
plane  of  the  ecliptic  at  the  time  of  conjunction  or  opposition, 
that  an  eclipse  of  the  sun  or  the  moon  is  possible. 

LUNAR  ECLIPSE 

150.  In  Fig.  57  let  S  be  the  center  of  the  sun,  and  E  that 
of  the  earth.     Draw  the  lines  B  H  and  CG,  tangent  to  the  two 
spheres.     These  lines  will  meet  at  some  point  A,  and  AH  EG 
will  be  a  section  of  the  shadow  cast  by  the  earth. 

The  whole  shadow  is  of  a  conical  shape,  the  vertex  of  the 

147 


148  LUNAR  AND  SOLAR  ECLIPSES 

cone  being  at  A;  and  a  lunar  eclipse  will  occur  whenever  the 
moon  is  within  this  shadow.  Draw  the  tangent  lines  BG 
and  C  H.  KDL  is  a  section  of  a  second  cone  whose  vertex  is 
at  D.  The  earth's  shadow  is  called  the  umbra,  and  that 
portion  of  the  second  cone  which  lies  outside  of  the  umbra  is 
called  the  penumbra.  Thus  KH A  and  AGL  are  sections  of 
the  penumbra.  It  must  be  noticed,  in  regard  to  the  construc- 
tion of  this  figure,  that  since  only  one  tangent  can  be  drawn 
to  the  circumference  of  a  circle  at  any  one  point,  the  lines 
BG  and  C  H  do  not  touch  the  two  circumferences  at  precisely 
the  same  points  at  which  BH  and  CG  touch;  and  that, 


M' 


FIG.  57. 

furthermore,  in  all  these  figures  the  relative  size  of  the  sun 
should  be  immensely  greater  than  it  is. 

Now  let  M'MM"  represent  a  portion  of  the  moon's  orbit 
at  the  time  of  a  lunar  eclipse.  As  soon  as  the  moon  passes 
within  the  line  DK,  some  of  the  rays  of  the  sun  will  be  cut 
off  from  it  by  the  earth,  and  its  brightness  will  begin  to 
decrease.  The  whole  disc,  however,  will  still  be  visible.  As 
soon  as  the  moon  begins  to  pass  within  the  line  HA ,  the  disc 
will  begin  to  disappear,  and  when  the  whole  disc  has  passed 
within  the  cone,  the  eclipse  will  be  total. 

151.  Different  Kinds  of  Eclipses.— When  the  moon's 
orbit  is  so  situated  that  only  a  part  of  the  moon  enters  the 
umbra,  we  have  a  partial  eclipse.  When  the  moon  does  not 
enter  the  umbra,  but  merely  touches  it,  we  have  an  appulse. 
When  the  center  of  the  moon  coincides  with  the  line  which 


LUNAR  ECLIPSE  149 

connects  the  center  of  the  earth  and  that  of  the  sun,  the 
eclipse  is  central  A  central  eclipse  occurs  very  rarely,  if 
indeed  it  occurs  at  all. 

152.  The  Semi-Angle  of  the  Umbral  Cone. — The  semi- 
angle  of  the  umbral  cone  is  the  angle  EAG,  Fig.  58.  Now  we 
have,  by  Geometry, 

SEC=ECG+EAG. 

But  SEC  is  the  sun's  angular  semi-diameter,  and  ECG 
is  its  horizontal  parallax.  Putting  S  for  SEC,  and  P  for 
ECG,  we  have, 

EAG=S-P. 


FIG.  58. 

153.  The  Angular  Semi-diameter  of  the  Shadow  at  the 
Distance  of  the  Moon.  —  The  angular  semi-diameter  of  the 
shadow  at  the  distance  of  the  moon  is  the  angle  MEM'.  We 
have,  by  Geometry, 

=  MEM'+EAG. 


Now  EM'G  is  the  moon's  horizontal  parallax,  which  we  will 
represent  by  P'  ',  and  the  value  of  EAG  has  been  obtained  in 
the  preceding  article.  We  therefore  have, 


Observation  shows  that  the  earth's  atmosphere  increases 
the  apparent  breadth  of  the  shadow  by  about  its  one-fiftieth 
part:  hence  in  practice  the  angular  semi-diameter  of  the 
shadow  is  taken  equal  to  fj  (P'+P—S).  If  we  substitute 


150 


LUNAR  AND  SOLAR  ECLIPSES 


in  this  expression  the  least  values  of  Pf  and  P,  and  the  great- 
est value  of  S,  from  the  table  given  in  Art.  155,  we  shall  find 
that  the  least  value  of  the  angular  semi-diameter  of  the 
shadow  is  about  37  25"  so  that  the  entire  breadth  of  the 
shadow  is  always  more  than  double  the  greatest  diameter  of 
the  moon. 

154.  Length  of  the  Earth's  Shadow.—  The  length  of  the 
shadow,  or  the  line  EA,  can  be  computed  from  the  right- 
angled  triangle  EAG  in  which  we  have, 

EA  =  EG  cosec  (S-P). 

The  mean  value  of  this  length  is  857,000  miles,  or  more 
than  three  times  the  distance  of  the  moon  from  the  earth. 

155.  Lunar  Ecliptic  Limits.  —  We  see  from  Fig.  58  that  a 
lunar  eclipse  can  occur  only  when  the  moon's  geocentric  lati- 
tude at  the  time  of  opposition  (or  at  full  moon),  is  less  than 
the  sum  of  the  angular  semi-diameter  of  the  shadow  and  the 
semi-diameter  of  the  moon.     If  we  represent  the  moon's 
semi-diameter  by  S'  the  expression  for  this  sum  is 


If  the  moon's  geocentric  latitude  at  the  time  of  opposition 
is  greater  than  the  greatest  value  which  this  expression  can 
attain,  no  eclipse  can  possibly  occur:  if  it  is  less  than  the  least 
value  of  the  expression,  an  eclipse  is  inevitable.  These  two 
values  of  this  expression  are  called  the  lunar  ecliptic  limits. 
Now,  we  have  by  observation  the  following  values  of  P,  P', 
etc.: 


Maxima. 

Minima. 

P' 

61'  32" 

52'  50" 

P 

9 

9 

S' 

16  46 

14  24 

S 

16   18 

15  45 

LUNAR  ECLIPSE  151 

In  order  to  find  the  greatest  value  of  the  expression,  we 
substitute  in  it  the  greatest  values  of  P,  Pr  and  S',  and  the 
least  value  of  S.  The  result  is  1°  3'  37":  and  no  eclipse  will 
occur  when  the  moon's  latitude  exceeds  this  limit.  The 
least  value  of  the  expression  is  51'  49" :  and  when  the  moon's 
latitude  at  opposition  is  less  than  this,  an  eclipse  cannot  fail 
to  occur.  There  are  some  considerations,  however,  which 
have  not  been  taken  into  account,  which  may  increase  each 
of  these  limits  by  about  16" . 

When  the  moon's  latitude  at  opposition  is  within  these 
limits,  an  eclipse  is  possible,  but  not  necessarily  certain.  In 
order  to  determine  whether  in  such  case  it  will  or  will  not 
occur,  the  actual  values  which  P,  P',  S  and  S'  will  have  at 
that  time  must  be  substituted  in  the  expression,  and  the 
result  compared  with  the  corresponding  latitude  of  the 
moon. 

156.  Since  a  lunar  eclipse  is  caused  by  the  moon's  enter- 
ing the  earth's  shadow,  it  will  be  seen  at  the  same  instant  of 
time  by  every  observer  who  has  the  moon  above  his  horizon : 
and  the  character  of  the  eclipse,  whether  total  or  partial  will 
be  everywhere  the  same.  As  the  moon's  motion  towards 
the  east  is  more  rapid  than  that  of  the  earth  (and  con- 
sequently of  the  shadow) ,  the  eclipse  will  begin  at  the  eastern 
limb  of  the  moon.  A  total  eclipse  of  the  moon  may  last  for 
nearly  two  hours.  Even  when  totally  eclipsed,  however, 
the  moon  does  not,  in  general,  disappear  from  view,  but 
shines  with  a  dull  reddish  light.  This  phenomenon  is 
caused  by  the  earth's  atmosphere,  which  refracts  the  rays  of 
light  from  the  sun  which  enter  it  near  the  points  G  and  H, 
Fig.  57,  and  turns  them  into  the  cone.  The  rays  which  pass 
still  nearer  to  these  points  are  probably  absorbed  by  the 
atmosphere,  thus  giving  rise  to  the  observed  increase  of  the 
shadow  mentioned  in  Art.  153. 


152  LUNAR  AND  SOLAR  ECLIPSES 


SOLAR  ECLIPSE 

157.  In  Fig.  59,  let  S  represent  the  sun,  E  the  earth,  and 
M  the  moon,  at  the  time  of  a  solar  eclipse.  HA  K  will  be  a 
section  of  the  moon's  umbra,  and  G  HA  and  A  KD,  sections 
of  its  penumbra. 

To  an  observer  situated  within  the  umbra,  at  any  point 
of  the  arc  ab,  the  eclipse  will  be  total;  while  to  one  situated 
within  the  penumbra,  as  at  L,  for  instance,  the  eclipse  will  be 
partial.  Beyond  the  penumbra  no  eclipse  whatever  will  be 
seen.  Hence  the  geographical  position  of  the  observer 


FIG.  59. 

determines  the  character  of  the  eclipse:  a  condition  different 
from  that  in  the  case  of  a  lunar  eclipse,  which  we  have  seen  is 
the  same  to  all  observers. 

158.  Length  of  the  Moon's  Shadow. — It  is  evident  that, 
to  an  observer  at  the  apex  of  the  shadow  A,  the  angular 
semi-diameters  of  the  sun  and  the  moon  would  be  equal. 
Now,  the  mean  angular  serm-diameters  of  these  two  bodies 
as  seen  from  the  earth's  center  are  nearly  equal;  hence  the 
mean  position  of  the  apex  does  not  fall  very  far  from  the 
earth's  center  E.  An  approximate  value  of  the  length  of 
the  shadow  may  be  thus  obtained,  We  have  in  Fig.  59, 

,,    HM    CS 

Sin  HAM**-T-fjr=*-r-s. 

AM    AS 


SOLAR  ECLIPSE  153 

But  we  have  just  now  seen  that  HAM  is  the  sun's 
angular  semi-diameter,  as  seen  from  A;  and  as  AE  is  small 
compared  with  AS,  we  may  consider  the  angle  HAM  to  be 
the  sun's  geocentric  semi-diameter.  Denoting  this  by  $, 
we  have, 

HM 


sin  S  = 


AM' 


Now,  if  S'-  represents  the  moon's  geocentric  semi-diameter, 
we  have, 

.     0,    HM 
= 


Combining  these  two  equations,  and  finding  the  value  of 
AM,  we  have, 


sin 

Knowing,  then,  the  distance  of  the  moon  from  the  earth's 
center,  and  the  semi-diameters  of  the  sun  and  the  moon,  we 
may  find  the  length  of  AM.  When  the  two  semi-diameters 
are  equal,  we  have  AM  equal  to  EM,  and  the  apex  is  at  the 
earth's  center.  When  the  semi-diameter  of  the  moon  is 
greater  than  that  of  the  sun,  the  apex  falls  beyond  the  earth's 
center:  when  it  is  less,  the  apex  does  not  reach  the  center. 
Appropriate  calculations  will  show  that  when  both  sun  and 
moon  are  at  their  mean  distances  from  us,  the  apex  falls 
short  of  the  earth's  surface:  and  that  when  the  moon  is  at 
its  least  distance  from  the  earth,  and  its  shadow  is  the  longest, 
the  apex  falls  about  14,000  miles  beyond  the  earth's  center. 
159.  Different  Kinds  of  Eclipses.  —  When  the  shadow  falls 
beyond  the  earth's  surface,  the  eclipse  is  total,  as  we  have 
already  seen,  within  the  umbra,  and  partial  within  the 
penumbra.  When  the  apex  just  touches  the  earth,  the 
eclipse  is  total  only  at  the  point  where  it  touches.  When 
the  apex  falls  short  of  the  surface,  there  will  be  no  total 


154  LUNAR  AND  SOLAR  ECLIPSES 

eclipse;  but  at  the  point  in  which  the  axis  of  the  cone,  pro- 
longed, meets  the  earth,  the  observer  will  see  what  is  called 
an  annular  eclipse,  the  moon  being  projected  upon  the  disc 
of  the  sun,  but  not  covering  it. 

160.  Solar  Ecliptic  Limits.— In  Fig.  60  let  S  represent 
the  sun,  E  the  earth,  and  M  the  moon.  No  eclipse  of  the 
sun  can  occur  unless  some  part  of  the  moon  passes  within  the 
lines  BC  and  GD,  drawn  tangent  to  the  sun  and  the  earth: 


FIG.  60. 


that  is,  unless  the  moon's  geocentric  latitude  is  less  than  the 
angle  MES.     Now  we  have, 


=  MEA+AEB+BES, 
and,  also, 

CAE-CBE. 


BES  is  the  sun's  semi-diameter,  ME  A  that  of  the  moon, 
CAE  the  moon's  horizontal  parallax,  and  CBE  the  sun's: 
hence,  using  the  notation  already  employed  in  Art.  155,  we 
have, 


.The  greatest  value  of  this  expression  is  found  by  employ- 
ing the  greatest  values  of  S,  S',  and  P',  and  the  least  value 
of  P,  as  given  in  Art.  155,  and  is  I'  34'  27":  and  there  will  be 
no  eclipse  if  the  moon's  latitude  at  conjunction  is  greater 
than  this  amount.  The  least  value  is  1°  22'  50";  and  if  the 
latitude  at  conjunction  is  less  than  this  an  eclipse  is  inevi- 


SOLAR  ECLIPSE  155 

table.  These  two  values,  which,  owing  to  certain  considera- 
tions omitted  in  this  discussion,  should  both  be  increased  by 
about  25' ',  are  called  the  solar  ecliptic  limits.  In  order  to 
determine  whether  an  eclipse  will  occur  when  the  moon's 
latitude  at  conjunction  falls  within  these  limits,  we  must 
substitute  in  the  expression  the  values  which  the  different 
quantities '  will  really  have  at  that  time,  and  compare  the 
result  with  the  corresponding  latitude  of  the  moon. 

161.  General  Phenomena. — Since  the  moon  moves 
towards  the  east  more  rapidly  than  the  sun,  a  solar  eclipse 
will  begin  at  the  western  side  of  the  sun.  For  the  same 
reason  the  moon's  shadow  will  cross  the  earth  from  west  to 
east,  and  the  eclipse  will  begin  earlier  at  the  western  portions 
of  the  earth's  surface  than  at  the  eastern.  The  moon's 
penumbra  is  tangent  to  the  earth's  surface  at  the  beginning 
and  the  end  of  the  eclipse,  so  that  the  sun  will  be  rising  at  that 
place  where  the  eclipse  is  first  seen,  and  setting  at  the  place 
where  it  is  last  seen.  A  solar  eclipse  may  last  at  the  equator 
about  4J  hours,  and  in  these  latitudes  about  3 J  hours.  That 
portion  of  the  eclipse,  however,  in  which  the  sun  is  wholly 
concealed  can  only  last  about  eight  minutes:  and  in  these 
latitudes,  only  about  six  minutes. 

The  darkness  during  a  total  eclipse,  though  subject  to 
some  variation,  is  scarcely  so  intense  as  might  be  expected. 
The  sky  often  assumes  a  dusky,  livid  color,  and  terrestrial 
objects  are  similarly  affected.  The  brighter  planets  and 
some  of  the  stars  of  the  first  magnitude  generally  become 
visible;  and  sometimes  stars  of  the  second  magnitude  are 
seen.  The  corona  and  the  rose-colored  protuberances  de- 
scribed in  Art.  102  also  make  their  appearance.  When  the 
sun's  disc  has  been  reduced  to  a  narrow  crescent,  it  some- 
times appears  as  a  succession  of  bright  points,  separated  by 
dark  spaces.  This  phenomenon  bears  the  name  of  B ally's 
beads.  The  dark  spaces  are  supposed  to  be  the  lunar  moun- 
tains, projected  upon  the  sun's  disc,  and  allowing  the  disc 
to  show  between  them. 


156  LUNAR  AND  SOLAR  ECLIPSES 

Occasionally  the  moon's  disc  is  faintly  seen,  shining  with  a 
dusky  light.  This  is  caused  by  the  rays  of  the  sun,  reflected 
back  to  the  moon  by  that  portion  of  the  earth's  surface 
which  is  still  illuminated  by  the  sun:  just  as  at  the  time  of 
new  moon  its  entire  disc  is  rendered  visible. 

CYCLE  AND  NUMBER  OF  ECLIPSES 

162.  Cycle  of  Eclipses. — In  order  that  either  a  solar  or  a 
lunar  eclipse  shall  occur,  it  is  necessary,  as  we  have  seen, 
that  the  moon  shall  be  near  the  ecliptic  (in  other  words,  near 
the  line  of  nodes  of  its  orbit),  at  either  conjunction  or  opposi- 
tion. It  is  evident  that  when  the  moon  is  near  the  line  of 
nodes  at  such  a  time  the  sun  also  must  be  near  the  same  line. 
The  occurrence  of  eclipses,  then,  depends  on  the  relative 
situations  of  the  sun,  the  moon,  and  the  moon's  nodes,  and 
is  only  possible  when  they  are  all  in,  or  nearly  in,  the  same 
straight  line.  We  have  already  seen  (Art.  128)  that  the  line 
of  nodes  is  continually  revolving  to  the  west,  completing 
a  revolution  in  about  18.6  years.  The  sun,  then,  in  its  appar- 
ent path  in  the  ecliptic  will  move  from  one  of  the  moon's 
nodes  to  the  same  node  again  in  less  than  a  year.  This 
interval  of  time  may  be  called  the  synodical  period .  of  the 
node,  and  is  found  to  be  346.62  days. 

Now,  we  have, 

19X346.62d.  =  6585.8d.; 

and,  the  lunar  month  being  29.53  days,  we  have  also, 
223  X  29.53d.  =  6585.2d. 

If,  then,  the  moon  is  full  and  at  its  node  on  any  day,  it  will 
again  be  full,  and  at  the  same  node,  or  very  nearly  at  it, 
after  an  interval  of  6585  days;  and  the  eclipses  which  have 
occurred  in  that  interval  will  occur  again  in  very  nearly  the 
same  order.  This  period  of  6585  days,  or  18  years  and  10 


OCCULTATIONS  157 

days,  is  called  the  cycle  of  eclipses.  It  was  known  to  the 
Chaldaean  astronomers  under  the  name  of  Saros.  Care  must 
be  taken  not  to  confound  this  cycle  with  the  lunar  cycle 
described  in  Art.  147. 

163.  Number  of  Eclipses. — Since  the  limit  of  the  moon's 
latitude  is  greater  in  the  case  of  a  solar  eclipse  than  in  the  case 
of  a  lunar  eclipse,  there  are  more  solar  eclipses  than  lunar 
eclipses.     Usually  70  eclipses  occur  in  a  cycle,  of  which  41 
are  solar  and  29  are  lunar.     Since  we  know  that  a  solar  eclipse 
is  inevitable  when  the  moon  is  so  near  the  line  of  nodes  at 
conjunction  that  its  latitude  is  less  than  1°  34'  27",  we  can 
compute  the  corresponding  angular  distance  of  the  sun  at 
the  same  time  from  this  line;  and  having  computed  this,  we 
may  also  determine  the  length  of  time  required  by  the  sun  in 
passing  through  double  this  angle,  or,  in  other  words,  the  time 
required  in  passing  from  one  of  these  limits  to  the  correspond- 
ing limit  on  the  other  side  of  the  same  node.     If  we  do  this, 
we  shall  find  that  the  sun  cannot  pass  either  node  of  the 
moon's  orbit  without  being  eclipsed:  and  therefore    there 
must  be  at  least  two  solar  eclipses  in  a  year.     The  greatest 
number  that  can  occur  is  five.     The  greatest  number  of 
lunar  eclipses  in  the  year  is  three,  and  there  may  be  none  at 
all.     The  greatest  number  of  both  kinds  of  eclipses  in  a  year 
is  seven;  the  usual  number  is  four. 

Although  the  annual  number  of  solar  eclipses  throughout 
the  whole  earth  is  the  greater  yet  at  any  one  place  more 
lunar  eclipses  are  visible  than  solar.  The  reason  of  this  is 
that  a  lunar  eclipse,  when  it  does  occur,  is  visible  over  an 
entire  hemisphere,  while  the  area  within  which  a  solar  eclipse 
is  visible  is  very  much  more  limited. 

OCCULTATIONS 

164.  An  occupation  of  a  planet  or  a  star  will  occur  when- 
ever the  planet  or  star  is  so  situated  in  latitude  as  to  allow 
the  moon  to  come  in  between  it  and  the  earth.     In  order  to 


158  LUNAR  AND  SOLAR  ECLIPSES 

determine  the  limit  of  a  planet's  latitude  within  which  an 
occultation  of  the  planet  is  possible,  let  us  refer  to  Fig.  61. 
In  this  figure,  E  is  the  center  of  the  earth,  P  that  of  a  planet, 
a*hd  M  that  of  the  moon.  An  occultation  will  occur  when  the 
moon  comes  between  the  tangent  lines  GB  and  A  H.  Let 


EC  be  the  plane  of  the  ecliptic.     PEC  is  then  the  geocentric 
latitude  of  the  planet,  and  MEC  that  of  the  moon, 
We  have, 


=  PEG+GED+DEM+MEC', 
and  also, 

GED  =  EDB-EGD. 

Now,  PEG  is  the  planet's  semi-diameter,  EGD  its  hori- 
zontal parallax,  DEM  the  moon's  semi-diameter,  EDB  its 
horizontal  parallax,  and  MEC,  as  above  stated,  its  latitude. 
The  value  of  PEC,  therefore,  can  very  readily  be  obtained. 
If  P,  instead  of  representing  a  planet,  represents  a  star,  the 
distance  PE  becomes  so  great  that  A  H  and  BG  are  sensibly 
parallel,  and  the  star's  parallax  and  semi-diameter  reduce  to 
zero.  In  this  case  the  greatest  value  of  PEC,  within  which 
an  occultation  can  occur,  will  be  the  sum  of  5°  20'  6",  61' 
32",  and  16'  46",  which  is  6°  38'  24". 


OCCULTATIONS  159 

Since  the  moon  moves  from  west  to  east,  the  occupation 
always  takes  place  at  its  eastern  limb.  From  new  moon  to 
full,  the  dark  portion  of  the  moon  is  to  the  east,  as  may  be 
seen  in  Fig.  54,  and  from  full  moon  to  new,  the  bright  limb  is 
to  the  east.  When  an  occultation  occurs  at  the  dark  edge, 
particularly  if  the  moon  is  so  far  on  towards  its  first  quarter 
that  the  dark  portion  is  invisible,  the  disappearance  is  ex- 
extremely  striking,  as  the  occulted  body  appears  to  be  extin- 
guished without  any  visible  interference. 

As  already  stated  in  Art.  83,  a  solar  eclipse,  or  an  occulta- 
tion of  a  planet  or  star,  although  not  visible  at  different 
places  at  the  same  absolute  instant  of  time,  may  still  be  made 
the  means  of  very  accurately  determining  the  longitude  of  a 
place,  or  the  difference  of  longitude  of  two  places.  For 
instance,  in  the  case  of  a  solar  eclipse,  we  may  deduce,  from 
the  local  times  of  the  beginning  and  the  end  of  the  eclipse,  as 
observed  at  any  place,  the  time  of  true  conjunction  of  the 
sun  and  the  moon:  the  time  of  conjunction,  that  is  to  say,  as 
seen  from  the  center  of  the  earth.  If,  then,  we  compare  the 
local  time  of  true  conjunction  with  the  Greenwich  time  of 
true  conjunction,  it  amounts  to  comparing  the  local  and  the 
Greenwich  time  corresponding  to  the  same  absolute  instant: 
and  the  difference  of  these  two  times  will  evidently  be  the 
longitude  of  the  place  of  observation  from  Greenwich. 


CHAPTER  XI 
THE  TIDES 

165.  THE  surface  of  the  ocean  rises  and  falls  twice  in  the 
course  of  a  lunar  day,  the  length  of  which  is,  as  we   have 
already  seen  (Art.  143),  about  24h.  50m.  of  mean  solar  time. 
When  the  water  is  at  its  greatest  height  it  is  said  to  be  high 
water  and,  when  at  its  least  height,,  low  water.     The  hori- 
zontal movement  of  the  water  is  called  a  tidal  current  and 
when  the  tide  is  rising  the  water  is  flowing  from  the  general 
direction  of  the  sea  to  the  land.     This  stage  is  called  flood 
tide  and  when  the  water  is  receding  from  the  land  the  tidal 
current  is  ebb. 

166.  Cause  of  the  Tides. — The  tides  are  due  to  the 
inequality  of  the  attractions  exerted  by  the  moon  upon  the 
earth  and  the  waters  of  the  ocean,  and  to  a  similar  but  smaller 
inequality  in  the  attractions  exerted  by  the  sun. 

In  order  to  examine  the  phenomena  of  the  tides  we  will 
consider  the  earth  to  be  a  solid  globe,  surrounded  by  a  shell 
of  water  of  uniform  depth.  The  centrifugal  force  induced  by 
the  rotation  of  the  earth  would  tend  to  give  a  spheroidal  form 
to  this  shell  of  water;  but  as  we  wish  simply  to  examine  the 
effects  of  the  attractions  exerted  by  the  moon  and  the  sun, 
we  will  disregard  the  "rotation  of  the  earth,  and  consider  it  to 
be  at  rest. 

In  Fig.  62,  then,  let  A  BCD  represent  the  earth,  and  the 
dotted  line  GHIK  the  surrounding  shell  of  water.  Let  M 
be  the  moon.  The  attraction  of  the  moon  on  the  solid  mass 
of  the  earth  is  the  same  that  it  would  be  if  the  whole  mass 
were  concentrated  at  the  point  E.  Now  since,  by  the  law  of 

160 


CAUSE  OF  THE  TIDES 


161 


gravitation,  the  attraction  of  the  moon  on  any  two  particles 
is  inversely  as  the  square  of  the  distances  of  the  two  particles 
from  the  moon,  the  attraction  exerted  upon  the  particle  of 
water  at  G  will  be  greater  than  that  exerted  upon  the  general 
mass  of  the  earth,  supposed  to  be  concentrated  at  E.  The 
particle  G  will  therefore  tend  to  recede  from  the  earth :  that  is 
to  say,  its  gravity  towards  the  earth's  center  will  be  dimin- 
ished, although,  as  is  plain,  it  will  not  move.  The  same 
result  will  follow  at  the  opposite  point  I.  The  moon  will 
exert  a  greater  attractive  power  upon  the  mass  of  the  earth 
than  upon  a  particle  at  J,  and  will  tend  to  draw  the  earth 


FIG.  62. 

away  from  the  particle:  so  that  the  gravity  of  the  particle 
at  /  towards  the  earth's  center  will  also  be  diminished.  Since 
the  ratio  of  the  distances  ME  and  MG  is  very  nearly  equal  to 
the  ratio  of  the  distances  MI  and  ME,  the  amount  of  the 
decrease  of  gravity  at  G  and  at  7  will  be  nearly  the  same. 
Let  us  next  examine  the  effect  of  the  moon's  attraction  at 
some  point  L,  not  situated  vertically  under  the  moon.  The 
attraction  of  the  moon  at  this  point  is  less  than  that  at  the 
point  G,  since  the  distance  ML  is  greater  than  the  distance 
MG;  and  since  the  attraction  exerted  on  the  mass  of  the 
earth  is,  of  course,  the  same  for  both  points,  the  difference 
of  the  attractions  exerted  on  the  earth  and  the  water  is  less 
at  the  point  L  than  at  the  point  G.  At  the  point  L,  however, 
this  inequality  of  attraction  is  not  wholly  counteracted  by 


162  THE  TIDES 

gravity:  for  if  the  force  with  which  the  moon  tends  to  draw  a 
particle  at  L  along  the  line  ML  be  resolved  into  two  forces, 
one  in  the  direction  of  the  radius  EL,  and  the  other  in  the 
direction  of  the  tangent  LT,  the  latter  force  will  cause  the 
particle  to  move  towards  the  point  G.  The  same  result  will 
follow  at  any  other  point  of  the  arc  HGK;  so  that  all  the 
water  in  that  arc  tends  to  flow  towards  the  point  G,  and  to 
produce  high  water  there. 

In  the  same  way  it  may  be  shown  that  the  water  in  the 
arc  HIK  tends  to  flow  towards  the  point  7,  and  to  produce 
high  water  at  that  point. 

The  result,  then,  of  the  attraction  of  the  moon,  exerted 
under  the  suppositions  which  we  made  at  the  outset,  is  to 
give  to  the  shell  of  water  a  spheroidal  form,  as  shown  in  the 
figure,  the  major  axis  of  the  spheroid  being  directed  towards 
the  moon.  Suitable  investigation  shows  that  the  difference 
of  the  major  and  the  minor  semi-axis  of  this  spheroid  is 
about  fifty-eight  inches. 

167.  Daily  Inequality  of  the  Tides.— The  rotation  of  the 
earth,  and  the  inclination  of  the  plane  of  the  moon's  orbit 
to  the  plane  of  the  equator,  produce  in  general  an  inequality 
in  the  two  daily  tides  at  any  place.  In  order  to  show  this, 
we  will  suppose  that  the  spheroidal  form  of  the  water  is 
assumed  instantaneously  in  each  new  position  of  the  earth 
as  it  rotates.  In  Fig.  63,  let  E  be  the  center  of  the  earth, 
surrounded,  as  in  Fig.  62,  by  a  spheroidal  shell  of  water,  the 
transverse  axis  of  the  sphercid  lying  in  the  direction  of  the 
moon  M .  Let  Pp  be  the  axis  of  rotation  of  the  earth,  and 
CD  the  equator.  The  angle  MED  is  the  moon's  declination. 
The  water  is  at  its  greatest  height,  as  before,  at  the  points 
A  and  B,  and  the  height  at  other  points  diminishes  as  the 
angular  distance  of  those  points  from  the  line  G  H  increases. 
Let  7  be  a  place  having  the  same  latitude  that  A  has,  but 
situated  180°  from  it  in  longitude.  The  height  of  the  tide 
at  7  is  represented  by  IK.  In  a  little  more  than  twelve 
hours  the  rotation  of  the  earth  will  have  caused  7  and  A  to 


DAILY  INEQUALITY  OF  THE  TIDES 


163 


change  places  with  reference  to  the  moon.  I  will  then  be 
where  A  is  in  the  figure,  and  will  have  a  tide  with  the  height 
AG,  while  A  will  be  where  /  is  now,  and  will  have  a  tide  with 
the  height  IK.  It  is  not  necessary  to  prove  that  IK  is  less 
than  AG.  We  see,  then,  that  at  both  A  and  I  the  two  daily 
tides  are  unequal,  the  greater  of  the  two  occurring  at  each 
place  at  the  time  of  the  moon's  upper  culmination  at  that 
place,  or  being,  at  all  events,  the  one  which  occurs  next  after 
that  culmination.  The  same  daily  inequality  of  tides  may 
be  shown  to  exist  at  any  other  points  on  the  earth's  surface, 


FIG.  63. 

as,  for  instance,  at  L  and  0.  At  the  equator,  however,  and 
also  at  the  poles,  the  two  daily  tides  are  sensibly  equal,  as 
may  readily  be  seen  from  the  figure. 

168.  General  Laws. — As  far  as  the  influence  of  the  moon 
is  concerned  in  causing  tides,  the  following  general  laws  may 
be  deduced  from  what  has  been  shown  in  the  preceding 
articles. 

(1)  When  the  moon  is  in  the  plane  of  the  celestial 
equator,  or,  in  Fig.  63j  when  EM  coincides  with  ED,  the 
tides  are  greatest  at  the  equator,  and  diminish  at  other  places 
as  the  latitude  increases;  and  the  two  daily  tides  at  any  place* 
are  sensibly  equal. 


164  THE  TIDES 

(2)  When  the  .moon  is  not  in  the  plane  of  the  celestial 
equator,  the  two  daily  tides  at  any  place  except  the  poles  and 
the  equator  are  unequal.    The  greatest  tides,  and  the  greatest 
inequality  of  tides,  occur  at  those  places  whose  latitude  is 
numerically  equal  to  the  moon's  declination.     If  the  place  is 
on  the  same  side  of  the  equator  as  the  moon,  the  greater  of  the 
two  daily  tides  occurs  at  or  next  after  the  upper  culmination 
of  the  moon ;  if  the  place  is  on  the  opposite  side  of  the  equator, 
the  greater  tide  occurs  at  or  next  after  the  lower  culmination 
of  the  moon. 

(3)  Owing  to  the  retardation  of  the  moon   (Art.   143), 
there  is  a  similar  retardation  in  the  occurrence  of  high  water. 
The  length  of  the  lunar  day  being  on  the  average  24h.  50m., 
the  average  interval  of  time  between  two  successive  tides  is 
12h.  25m. 

169.  Influence  of  the  Sun  in  Causing  Tides. — All  that 
has  been  said  in  the  preceding  articles  with  regard  to  the 
influence  of  the  moon  in  creating  tides  is  equally  true  with 
regard  to  the  influence  of  the  sun.     The  mass  of  the  sun 
being  so  immense  in  comparison  with  that  of  the  moon,  it 
might  be  supposed  that  the  influence  of  the  sun  over  the 
tides  would  be  greater  than  that  of  the  moon,  even  although 
its  distance  from  the  earth  is  much  greater  than  that  of  the 
moon.     But  such  is  not  the  case  in  fact.     The  height  of  the 
tide  produced  by  either  body  is  not  so  much  due  to  the  abso- 
lute attraction  which  that  body  exerts,  as  to  the  relative 
attractions  which  it  exerts  on  the  solid  mass  of  the  earth  and 
on  the  water:  and  the  moon  is  so  much  nearer  to  the  earth 
than  the  sun,  that  the  difference  of  its  attractions  on  the 
earth  and  the  water  is  greater  than  the  corresponding  dif- 
ference in  the  case  of  the  sun.     It  is  computed  that  the  effect 
of  the  moon  in  creating  tides  is  about  2|  times  that  of  the 
sun. 

170.  Combined  Effects  of  both  Sun  and  Moon. — Since 
each  body,  independently  of  the  other,  tends  to  raise  the 
surface  of  the  water  at  certain  points,  and  to  depress  it  at 


PRIMING  AND  LAGGING  OF  THE  TIDES  163 

certain  other  points,  the  tides  will  evidently  be  higher  when 
both  bodies  tend  to  raise  the  surface  of  the  water  at  the  same 
time,  than  when  one  tends  to  raise  and  the  other  to  depress 
it.  At  new  and  full  moon  the  two  bodies  act  together,  while 
at  the  first  and  the  last  quarter  they  act  in  opposition  to  each 
other.  The  tides  at  the  former  periods  will  therefore  be  the 
greater,  and  are  called  spring  tides;  and  the  tides  at  the 
latter  periods  are  called  neap  tides.  The  ratio  of  the  spring 
to  the  neap  tide  is  that  of  (2J+1)  to  (2J  —  1),  or  of  5  to  2. 

The  height  of  the  tide  is  also  affected  by  the  change  in  the 
distance  of  the  attracting  body.  For  instance,  when  the 
moon  is  in  perigee,  the  tides  tend  to  run  higher  than  when  the 
moon  is  in  apogee ;  and  when  the  moon  is  in  perigee,  and  also 
either  new  or  full,  unusually  high  tides  will  occur. 

171.  Priming  and  Lagging  of  the  Tides. — Each  of  these 
bodies  may  be  supposed  to  raise  a  tidal  wave  of  its  own,  and 
the  actual  high  water  at  any  place  may  be  considered  to  be 
the  result  of  the  combination  of  the  two  waves.     When  the 
moon  is  in  its  first  or  its  third  quarter,  the  solar  wave  is  to 
the  west  of  the  lunar  one,  and  the  actual  high  water  will  be 
to  the  west  of  the  place  at  which  it  would  have  been  had  the 
moon  acted  alone.     There  is  therefore  at  these  times  an 
acceleration  of  the  time  of  high  water,  which  is  called  the 
priming  of  the  tides.     In  the  second  and  the  fourth  quarter, 
the  solar  wave  is  to  the  east  of  the  lunar  one,  and  a  retarda- 
tion of  the  time  of  high  water  occurs,  which  is  called  the 
lagging  of  the  tides. 

172.  Although  the  theory  of  the  tides,  on  the  supposition 
that  the  earth  is  wholly  covered  with  water,  admits  of  easy 
explanation,  the  actual  phenomena  which  they  present  are 
very  much  more  complicated,  and  must  be  obtained  prin- 
cipally from  observation.     The  lunar  wave  mentioned  in 
the  preceding  article  being  greater  than  the  solar  wave,  we 
may  consider  the  two  together  to  constitute  one  great  tidal 
wave,  which  at  every  moment  tends  to  accompany  the  moon 
in  its  apparent  diurnal  path  towards  the  west,  raising  the 


166  THE  TIDES 

waters  at  successive  meridians,  but  giving  them  little  or  no 
progressive  motion.  This  tidal  wave  would  naturally  move 
westward  with  an  angular  velocity  equal  to  that  of  the  moon, 
so  that  at  the  equator  its  motion  would  be  about  1000  miles 
an  hour;  but  the  obstructions  offered  by  the  continents, 
the  irregularity  of  their  outlines,  the  uneven  surface  of  the 
ocean  bed,  and  the  action  of  winds  and  currents  and  friction, 
all  combine  not  only  to  diminish  the  velocity  of  the  tidal . 
wave,  but  also  to  make  it  extremely  variable. 

173.  Establishment  of  a  Port. — The  interval  of  time 
between  the  moon's  transit  over  any  meridian  and  high 
water  at  that  meridian  varies  at  different  places,  and  varies 
also  on  different  days  at  the  same  place.  This  interval  of 
time  is  called  the  luni-tidal  interval.  The  mean  of  the 
values  of  this  interval  on  the  days  of  new  and  full  moon  is 
called  the  common  establishment  of  a  port.  The  mean  of  all 
the  luni-tidal  intervals  in  the  course  of  the  month  is  called 
the  corrected  establishment.  These  establishments  are 
obtained  by  observation,  and  are  given  in  Bowditch's 
Navigator,  and  also  in  other  works.  Thus  the  establish- 
ment of  Annapolis  is  4h.  39  m.,  and  that  of  Boston  llh  27m. 
The  time  of  transit  of  the  moon  over  any  meridian  on  any 
day  can  be  obtained  from  the  Nautical  Almanac:  and  the 
sum  of  this  time  of  transit  and  of  the  establishment  of  any 
port  will  be  the  approximate  time  of  that  high  tide  which 
occurs  next  after  the  transit.  Suppose,  for  instance,  the 
time  of  high  water  at  Annapolis,  on  January  3,  1919,  is 
desired.  We  have  from  the  Almanac  the  time  of  the  moon's 
transit,  Ih.  20m.;  adding  to  this  the  establishment,  4h.  39m., 
the  sum  is  5h.  59m.  This  is,  in  this  case,  the  time  of  the 
evening  tide.  The  time  of  the  morning  tide  may  be  obtained 
by  subtracting  12h.  25m.  from  this  time,  which  will  give  us 
5h.  34m.  A.M.  A  more  accurate  result  in  this  last  case 
might  be  obtained  by  taking  from  the  Almanac  the  time  of 
the  preceding  lower  transit,  and  adding  the  establishment 
to  it;  but  practically  this  would  be  a  needless  refine- 


COTIDAL  TIDES  167 

ment,  for  the  two  results  would  vary  by  only  about  two 
minutes. 

The  time  of  transit  which  the  Almanac  gives  is  in  astro- 
nomical time :  hence  the  resulting  time  of  high  water  will  also 
be  in  astronomical  time,  and  it  will  frequently  happen  that 
the  time  which  we  find,  when  turned  into  civil  time,  will  fall 
on  the  civil  day  subsequent  to  the  day  for  which  the  time  is 
desired.  Take  for  instance,  April  23,  1919,  at  Annapolis. 
The  time  of  transit  is  April  23,  19h.  07m.:  hence  the  time  of 
high  water  is  April  23,  23h.  46m.,  or,  in  civil  time,  April  24, 
llh.  46m.  A.M.  If,  then,  we  wish  the  time  of  high  water  on 
the  morning  of  the  civil  day  April  23,  we  must  take  from  the 
Almanac  the  time  of  transit  for  the  astronomical  day  of 
April  22. 

The  United  States  Coast  Survey  publishes  annually,  in 
advance,  tables  giving  for  every  day  in  the  year  the  pre- 
dicted times  and  heights  of  high  and  low  waters  at  the 
principal  ports  in  the  world,  and  from  these,  by  a  simple 
reduction  explained  in  the  Tide  Tables,  the  times  and  heights 
at  other  seaports  may  readily  be  obtained. 

174,  Cotidal  Lines.— If  the  tidal  wave  were  everywhere 
uniform  in  its  progress,  it  would  come  to  all  points  on  the 
same  meridian  at  the  same  time.     But,  owing  to  irregularities 
induced  by  local  causes,  such  is  not  the  case,  and  places  on 
different  meridians  often  have  high  water    at    the    same 
instant  of  time.     Charts  are  therefore  published  on  which 
are  drawn  lines  connecting  places  where  high  tides  occur  at 
the  same  instant:  and  these  lines  are  called  cotidal  lines. 
These  lines  are  usually  accompanied   by   numerals,  which 
indicate  the  hours  of  Greenwich  time  at  which  high  tides 
occur  on  the  days  of  new  and  full  moon  along  the  different 
lines. 

175.  Height  of  Tides. — At  small  islands  in  mid-ocean  the 
height  of  the  tides  is  not  great,  being  sometimes  less  than 
one  foot.     When  the  tidal  wave  approaches  a  continent, 
and  the  water  begins  to  shoal,  the  velocity  of  the  wave  is 


168  THE  TIDES 

diminished,  and  the  height  of  the  tide  is  increased.  When 
the  wave  enters  bays  opening  in  the  direction  in  which  the 
wave  is  moving,  the  height  of  the  tide  is  still  further  in- 
creased. 

The  eastern  coast  of  the  United  States  may  be  considered 
to  constitute  three  great  bays:  the  first  included  between 
Cape  Sable,  in  Nova  Scotia,  and  Nantucket,  the  second 
between  Nantucket  and  Cape  Hatteras,  and  the  third 
between  Cape  Hatteras  \  and  Cape  Sable,  in  Florida.  In 
each  of  these  bays  the  tides,  in  general,  increase  in  height 
from  the  entrance  of  the  bay  to  its  head.  For  instance,  in 
the  most  southern  of  these  bays,  the  tides  at  Cape  Sable 
and  Cape  Hatteras  are  not  more  than  two  feet  in  height; 
while  at  Port  Royal,  at  the  head  of  this  bay,  they  are  about 
seven  feet.  The  same  thing  is  noticed,  in  general,  in  smaller 
bays  and  sounds.  For  instance,  in  Long  Island  Sound,  the 
height  of  the  tide  is  two  feet  at  the  eastern  extremity,  and 
more  than  seven  feet  at  the  western.  This  increase  of  height 
is  particularly  noticeable  in  the  Bay  of  Fundy,  in  which  the 
height  is  eighteen  feet  at  the  entrance,  and  fifty  and  some- 
times seventy  feet  at  the  head. 

There  are  in  some  cases,  however,  special  causes  which 
create  exceptions  to  this  general  rule  of  increase  of  the  tides 
between  the  entrance  and  the  head  of  a  bay.  In  Chesapeake 
Bay,  for  instance,  which  is  wider  at  some  places  than 
it  is  at  the  entrance,  and  which  lies  about  north  and 
south,  the  tides  in  general  diminish  in  height  as  we  ascend 
the  bay. 

176.  Tides  in  Rivers. — The  same  general  principle  holds 
good  in  the  tides  of  rivers.  When  the  channel  contracts  or 
shoals  rapidly,  the  height  of  the  tide  increases:  when  it 
widens  or  deepens,  the  height  decreases.  In  a  long  river, 
then,  the  tides  may  alternately  increase  and  decrease.  For 
instance,  at  Tivoli,  on  the  Hudson,  between  West  Point  and 
Albany,  the  tide  is  higher  than  it  is  at  either  of  those  two 
places. 


DIFFERENT  DIRECTIONS  OF  THE  TIDAL  WAVE         169 

177.  Different    Directions    of    the    Tidal    Wave.— The 

tidal  wave  naturally  tends  to  move  towards  the  west;  but, 
the  obstructions  offered  by  the  continents  and  the  promon- 
tories, and  the  irregular  conformation  of  the  bottom  of  the 
ocean,  materially  change  the  direction  of  its  motion.  Some- 
times its  direction  is  even  towards  the  east.  From  a  point 
about  one  thousand  miles  southwest  of  South  .America  there 
appear  to  start  two  tidal  waves,  which  travel  in  nearly 
opposite  directions,  one  towards  the  west  and  the  other 
towards  the  east. 

178.  Four  Daily  Tides. — At  some  places  the  tides  rise  and 
fall  four  times  in  each  day.     This  is  ascribed  to  the  existence 
of  two  different  tidal  waves,  coming  from  opposite  directions. 
This  phenomenon  occurs  on  the  eastern  coast  of  Scotland, 
where  one  wave  comes  into  the  North  Sea  through  the 
English  Channel,  while  a  second  wave  comes  in  around  the 
northern  extremity  of  Scotland.     At  places  where  these  two 
waves  arrive  at  different  times,  each  wave  will  produce  two 
daily  tides. 

179.  Tides  in  Lakes  and  Inland  Seas. — If  there  is  any  tide 
in  lakes  and  inland  seas,  it  is  usually  so  slight  as  to  be  scarcely 
measurable.     A  series  of  careful  observations  has  demon- 
strated the  existence  of  a  tide  in  Lake  Michigan,  which  is  at 
its  height  about  half  an  hour  after  the  moon's  transit.     The 
average  height  which  it  attains,  however,  is  less  than  two 
inches. 

180.  Other  Phenomena. — Along  the  northern  coast  of 
the  Gulf  of  Mexico  there  is  only  one  tide  in  the  day,  the 
second  one  being  probably  obliterated  by  the  interference  of 
two  waves.     An  approximation  to  this  state  of  things  is 
noticed  on  the  Pacific  coast,  where  at  times  one  of  the  daily 
tides  has  a  height  of  several  feet,  and  the  other  a  height  of 
only  a  few  inches.     A  very  curious  statement  has  been  made 
by  missionaries  concerning  the  tides  of  the  Society  Islands. 
They  say  that  the  tides  there  are  uniform,  not  only  in  the 
height  which  they  attain,  but  in  the  time  of  ebb  and  flow, 


170  THE  TIDES 

high  tide  occurring  invariably  at  noon  and  at  midnight;  so 
that  the  natives  distinguish  the  hour  of  the  day  by  terms 
descriptive  of  the  condition  of  the  tide. 

NOTE. — It  is  now  generally  admitted  that  one  result  of  the  friction 
of  the  tides  is  a  diminution  of  the  velocity  of  the  earth's  rotation;  and 
it  is  possible  that  the  moon's  secular  acceleration  (page  144)  is  partly 
due,  not  to  an  increase  in  the  moon's  orbital  velocity,  but  to  this  same 
diminution  of  the  earth's  rotation.  The  amount  of  the  diminution  is, 
however,  so  very  small  that  all  attempts  to  compute  it  have  been  thus 
far  unsuccessful. 


CHAPTER  XII 

THE   PLANETS   AND   THE   PLANETOIDS.    THE   NEBULAR 
HYPOTHESIS 

181.  The  Planets  and  Their  Apparent  Motions. — There 
are  other  celestial  bodies  besides  the  sun  and  the  moon, 
which,  while  they  share  the  common  diurnal  motion  towards 
the  west,  appear  to  change  their  relative  positions  among  the 
stars.     These  bodies  are  called  planets,  from  a  Greek  word 
signifying  wanderer.     Some  of  them  are  visible  to  the  naked 
eye,  and  some  only  become  visible  by  the  aid  of  a  telescope. 
In  some  of  them  the  change  of  position  among  the  stars 
becomes  apparent  from  the  observations  of  a  few  nights: 
while  in  others  even  the  annual  change  of  position  is  very 
small.     It  was  not  change  of  position  which  led  to  the  dis- 
covery of  Uranus  and  Neptune. 

This  change  of  position  is  determined,  as  it  was  deter- 
mined in  the  case  of  the  sun  and  the  moon,  by  observations 
of  their  right  ascensions  and  declinations.  When  such 
observations  are  made,  the  apparent  motions  of  the  planets 
are  found  to  be  very  irregular.  Sometimes  they  appear  to 
move  towards  the  east,  and  sometimes  towards  the  west, 
while  at  other  times  they  appear  to  be  stationary  in  the 
heavens.  Such  irregularity  in  the  direction  of  their  motion 
is  at  once  seen  to  be  incompatible  with  the  supposition 
that  they  are,  like  the  moon,  satellites  of  the  earth,  revolving 
about  it  as  a  center.  The  next  supposition  that  will  naturally 
be  made  is  that  they  may  revolve  about  the  sun. 

182.  Heliocentric  Parallax. — In  order  to  test  the  correct- 
ness of  this  second  supposition,  we  must  first,  from  the 

171 


172  THE  PLANETS  AND  THE  PLANETOIDS 

apparent  motions  which  are  observed  from  the  earth,  deduce 
the  corresponding  motions  which  would  be  seen  by  an  ob- 
server stationed  at  the  sun.  Fig,  64  will  serve  to  show  how,  by 
means  of  a  body's  heliocentric  parallax  (Art.  56),  the  position 
which  that  body  would  have  if  seen  from  the  sun's  center — 
in  other  words,  its  heliocentric  position — may  be  determined 
from  its  geocentric  position.  In  this  figure  let  S  represent 
the  sun,  ABCE  the  earth's  orbit  the  plane  of  which  intersects 
the  celestial  sphere  in  the  circle  VHD,  and  P  the  position  of 


a  planet  when  projected  upon  the  plane  of  the  ecliptic.  Let 
the  vernal  equinox  be  supposed  to  lie  in  the  direction  EV  or 
SV,  which  two  lines  must  be  supposed  sensibly  to  meet 
when  prolonged  to  the  celestial  sphere.  Draw  the  lines 
EP  and  SP.  Since  the  distances  of  the  planet  from  the 
sun  and  the  earth  are  finite,  these  lines  will  not  lie  in  the 
same  direction,  and  the  angle  EPS  which  they  make  with 
each  other  will  be  the  difference  of  the  directions  in  which  the 
planet  is  seen  from  the  earth  and  the  sun :  in  other  words,  the 
planet's  heliocentric  parallax  in  longitude.  Through  S 


ORBITS  OF  THE  PLANETS  173 

draw  SK  parallel  to  ED.     The  angle  VEP,  or  its  equal, 
VSK,  is  the  planet's  geocentric  longitude,  and  is  obtained  by 
observation.     The  sum  of  this  angle  and  the  angle  EPS 
is  the  angle   VSP,  or  the  planet's  heliocentric  longitude. 
Provided,  then,  we  know  the  angle  EPS,  we  can  readily 
obtain  the  angle  VSP.     Now,  in  the  triangle  PES  we  know 
from  Kepler's  Third  Law  the  ratio  of  the  sides  SP  and  ES, 
which  are  the  distances  of  the  planet  and  the  earth  from  the 
sun;  and  the  angle  PES,  the  planet's  angular  distance  from 
the  sun,  or  its  elongation  (Art.  56),  can  be  obtained  by 
-  observation.     The  angle  EPS  may  then  be  readily  computed. 
By  a  similar  method  the  planet's  heliocentric  latitude  may 
be  determined  from  its  geocentric  latitude,  and  the  helio- 
centric place  of  a  planet  may  thus  be  obtained  at  any  time. 

183.  Orbits  of  the  Planets,— When  the  motions  of  the 
planets  as  they  would  be  seen  from  the  center  of  the  sun 
are  thus  deduced  from  their  observed  motions  with  refeiv 
ence  to  the  earth,  all  the  apparent  irregularities  of  motion 
disappear.     The  planets  are  found  to  revolve  from  west  to 
east  in  ellipses  about  the  sun  in  one  of  the  foci,  the  eccentric- 
ity of  the  ellipses  diminishing,  as  a  general    rule,  as    the 
magnitude  increases.     The  planes  of  the  orbits  are  found  to 
be  nearly  coincident  with  the  plane  of  the  ecliptic,  and 
Kepler's  and  Newton's  laws  are  exactly  fulfilled  in  the  case 
of  each  planet.     The  line  in  which  the  plane  of  each  planet 
intersects  the  plane  of  the  ecliptic  is  called  the  line  of  nodes, 
and  the  terms  perihelion  and  aphelion  have  the  same  signi- 
fication that  they  have  in  the  case  of  the  earth. 

184.  Inferior  and  Superior  Planets. — The  planets  are 
divided  into  two  classes:  inferior  and  superior  planets.     The 
inferior  planets  are  those  whose  distances  from  the  sun  are 
less  than  the  distance  of  the  earth  from  the  sun,  and  whose 
orbits  are  therefore  included  within  the  orbit  of  the  earth. 
The  inferior  planets  are  Mercury  and  Venus.     The  superior 
planets  are  those  whose  distances  from  the  sun  are  greater 
than  that  of  the  earth,  and  whose  orbits  therefore  include 


174  THE  PLANETS  AND  THE  PLANETOIDS 

the  orbit  of  the  earth.  The  superior  planets  are  Mars, 
Jupiter,  Saturn,  Uranus,  and  Neptune.  There  is  besides 
these  a  group  of  small  planets,  called  minor  planets,  planetoids, 
or  asteroids,  situated  between  Mars  and  Jupiter.  Up  to 
1916,  over  800  of  these  minor  planets  had  been  discovered. 
The  earth  is  also  a  planet,  lying  between  Venus  and  Mars. 
It  is  therefore  a  superior  planet  to  Mercury  and  Venus,  and 
an  inferior  planet  to  the  other  planets.  Its  sidereal  period 
is  greater  than  the  periods  of  the  inferior  planets,  and  less 
than  those  of  the  superior  planets. 

INFERIOR  PLANETS 

185.  In  Fig.  65  let  8  represent  the  sun,  pp'p"pff"  the 
orbit  of  an  inferior  planet,  the  plane  of  which  is  supposed 
to  coincide  with  the  plane  of  the  ecliptic,  ABDE  the  orbit  of 
the  earth,  and  the  circle  KGH  the  intersection  of  the  plane 
of  the  ecliptic  with  the  celestial  sphere.     Suppose  the  earth 
to  be  at  E.    When  the  planet  is  at  P,  between  the  earth 
and  the  sun,  or  at  P'n ',  on  the  opposite  side  of  the  sun  to  the 
earth,  it  has  the  same  geocentric  longitude  as  the  sun,  and  is 
in  conjunction  with  it.    The  position  at  P  is  called  the  in- 
ferior conjunction,  and  that  at  P"'  the  superior  conjunction. 

The  greatest  angular  distance  of  the  planet  from  the  sun 
will  evidently  occur  when  the  line  connecting  the  planet  and 
the  earth  is  tangent  to  the  orbit  of  the  planet;  that  is  to  say, 
when  the  planet  is  at  P"  or  P"".  The  position  at  P"  is 
called  the  greatest  western  elongation  of  the  planet,  that  at 
P""  the  greatest  eastern  elongation  of  the  planet.  We  have 
already  seen  in  Art.  92  that  the  relative  distances  of  the 
planet  and  the  earth  from  the  sun  are  at  once  obtained 
when  we  have  measured  the  greatest  elongation. 

186.  Direct  and  Retrograde  Motion. — The  motion  of  the 
inferior  planets  is  always  in  reality  from  west  to  east,  or 
direct,  as  it  is  called;  but  when  the  planet  is  near  its  inferior 
conjunction,  its  motion  is  apparently  from  east  to  west,  or 


INFERIOR  PLANETS 


175 


retrograde.  This  apparent  retrograde  motion  is  explained  in 
Fig.  65.  Let  the  planet  be  at  its  inferior  conjunction  at  P, 
and  let  both  the  earth  and  the  planet  move  on  about  the  sun 
in  the  direction  EABD.  The  angular  and  the  linear  velocity 
of  the  planet  about  the  sun  being  greater  than  they  are  in 
the  case  of  the  earth,  when  the  earth  arrives  at  E' ',  the  planet 
will  be  at  some  point  P1 ',  and  will  lie  in  the  direction  E'G; 
the  sun,  on  the  other  hand,  will  lie  in  the  direction  E'S'. 


FIG.  65. 

While,  then,  the  earth  is  advancing  from  E  to  Ef,  the  sun 
and  the  planet  appear  to  move  away  in  opposite  directions 
from  the  point  C,  on  which  both  were  projected  when  the 
earth  was  at  E.  But  the  apparent  motion  of  the  sun  is  in- 
variably towards  the  east;  hence  the  planet  has  apparently 
moved  towards  the  west.  It  may  also  be  shown  that  the 
same  apparent  retrograde  motion  occurs  when  the  planet  is 
approaching  inferior  conjunction,  and  is  within  a  short 
distance  of  it. 

187.  Stationary  Points.— When  the  planet  is  at  P"n ', 


176  THE  PLANETS  AND  THE  PLANETOIDS 

y 

it  is  moving  directly  towards  the  earth  in  the  direction 
P""E,  and  the  motion  of  the  earth  in  its  orbit  gives  the 
planet  an  apparent  motion  in  advance.  The  same  must  also 
be  the  case  when  the  planet  is  at  P" .  Since  then  the  motion 
of  the  planet  is  direct  at  the  greatest  elongations,  and  retro- 
grade at  inferior  conjunction,  there  must  be  a  point  in  the 
orbit  between  inferior  conjunction  and  each  elongation  at 
which  the  planet  neither  advances  nor  recedes,  but  appears 
stationary  in  the  heavens.  These  points  are  called  the 
stationary  points. 

At  all  other  parts  of  the  orbit  except  those  which  have 
been  discussed,  the  apparent  motion  of  the  planet  is  direct; 
but  the  velocity  with  which  it  moves  is  subject  to  great 
variation.  It  was  on  account  of  this  irregularity,  both  in 
the  direction  and  the  amount  of  their  apparent  motions, 
that  these  bodies  were  called  wandering  stars  by  the  ancient 
Greeks. 

188.  Evening  and  Morning  Stars. — Except  at  the  times 
of  conjunction,  an  inferior  planet  is  either  to  the  east  or  to 
the  west  of  the  sun.     When  it  is  to  the  east  of  the  sun  it  will 
set  after  the  sun  has  set,  and  when  it  is  to  the  west  of  the 
sun  it  will  rise  before  the  sun  has  risen.     In  certain  parts  of 
its  orbit  the  planet's  elongation  from  the  sun  is  sufficiently 
great  to  carry  the  planet  beyond  the  limits  of  twilight,  or 
18°  (Art.  101) ;  it  will  then  be  an  evening  star  if  to  the  east  of 
the  sun,  and  a  morning  star  if  to  the  west.     It  is  only  to 
Venus,  however,  that  these  terms  are  commonly  applied,  at 
least  so  far  as  the  inferior  planets  are  concerned:  since 
Mercury  is  so  near  to  the  sun  that  it  is  seldom  visible,  and 
even  when  it  is  visible,  it  appears  like  a  star  of  only  the 
third  or  the  fourth  magnitude. 

189.  Elements  of  a  Planet's  Orbit. — In  order  to  compute 
the  position  in  space  at  any  time  of  either  an  inferior  or  a 
superior  planet,  we  must  be  able  to  determine : 

1st.  The  relative  position  of  the  plane  of  the  planet's 
orbit  to  the  plane  of  the  ecliptic;  "^ 


ELEMENTS  OF  A  PLANET'S  ORBIT  177 

2d.  The  position  of  the  orbit  itself  in  the  plane  in  which 
it  lies; 

3d.  The  magnitude  and  the  form  of  the  orbit;  and 

4th.  The  position  of  the  planet  in  its  orbit. 

These  four  conditions  require  the  knowledge  of  seven 
distinct  quantities.  The  first  condition  is  satisfied  if  we 
know  (1)  the  position  of  the  line  in  which  the  plane  of  the 
orbit  intersects  the  plane  of  the  ecliptic,  or,  what  amounts  to 
the  same  thing,  the  longitude  of  the  planet's  nodes,  and  (2) 
the  inclination  of  the  two  planes  to  each  other.  The  second 
condition  is  satisfied  if  we  know  (3)  the  longitude  of  the 
perihelion.  The  third  condition  is  satisfied  if  we  know  (4) 
the  semi-major  axis,  or  the  planet's  mean  distance  from  the 
sun,  and  (5)  the  eccentricity  of  the  orbit.  Finally,  the 
last  condition  is  satisfied  if  we  know  (6)  the  time  in  which  it 
makes  one  complete  revolution  about  the  sun,  or  its  periodic 
time,  and  (7)  the  time  when  the  planet  is  at  some  known  place 
in  its  orbit,  as  for  instance,  the  perihelion.  These  seven 
quantities  are  called  the  elements  of  the  orbit. 

190.  Heliocentric  Longitude  of  the  Node. — A  planet  is 
at  its  nodes  when  its  latitude  is  zero;  and  if  the  heliocentric 
longitude  of  the  planet  at  that  time  can  be  determined,  it 
will  also  be  the  heliocentric  longitude  of  the  node,  since  the 
line  of  nodes  of  every  planet  passes  through  the  sun.  But 
the  heliocentric  longitude  of  a  planet  when  at  its  node  differs 
from  the  geocentric  longitude  (which  may  be  obtained 
directly  from  observation),  excepting  only  in  case  the  earth 
itself  happens  at  that  time  to  be  on  the  line  of  nodes.  We 
must  therefore  be  able  to  deduce  the  heliocentric  longitude 
of  the  planet  from  the  geocentric.  When  the  planet's 
distance  from  the  sun  is  known,  this  can  be  done  by  the 
method  explained  in  Art.  182;  and  when  this  distance  is  not 
known,  the  following  method  can  be  used. 

In  Fig.  66,  let  S  be  the  sun,  PGHK  the  orbit  of  a  planet, 
CDEE'  the  orbit  of  the  earth,  and  NM  the  line  of  nodes  of 
the  planet.  Let  the  vernal  equinox  lie  in  the  direction  EV 


178 


THE  PLANETS  AND  THE  PLANETOIDS 


or  SY*.  Let  E  be  the  position  of  the  earth  when  the  planet 
is  on  the  line  of  nodes  at  P.  The  elongation  of  the  planet,  or 
the  angle  PES,  and  the  geocentric  longitude  of  both  planet 
and  node,  or  the  angle  VEP,  can  be  obtained  by  observation. 
Suppose  both  planet  and  earth  to  move  on  in  their  orbits, 
and  the  earth  to  be  at  E'  when  the  planet  again  reaches  the 
same  node,  and  let  the  planet's  elongation  at  this  time,  or  the 


angle  SE'P,  be  observed.  Now,  since  the  earth's  orbit, 
although  represented  in  the  figure  by  a  circle,  is  really  an 
ellipse,  ES  and  E'S  will  not  in  general  be  equal  to  each  other. 
The  value  of  each,  however,  can  be  readily  obtained  from  the 
solar  tables.  The  same  tables  will  also  give  us  the  angle 
ESEf,  which  is  the  angular  advance  of  the  earth  in  its  orbit 
in  the  interval.  In  the  triangle  ESE',  then,  knowing  two 
sides  and  the  included  angle,  we  can  compute  the  side  EE', 
and  the  angles  SEE'  and  SE'E.  The  angles  PES  and  PE'S 


INCLINATION  OF  PLANET'S  ORBIT  TO  ECLIPTIC    179 

having  been  obtained  by  observation,  we  can  find  the  angles 
PEE'  and  PE'E.  Then  in  the  triangle  PEE',  knowing 
two  angles  and  the  included  side,  we  can  compute  the  side 
EP.  Finally,  in  the  triangle  PES,  knowing  an  angle  and 
the  two  including  sides,  we  can  obtain  PS,  or  the  planet's 
distance  from  the  sun,  and  the  angle  EPS,  the  planet's 
heliocentric  parallax,  from  which,  and  the  planet's  observed 
geocentric  longitude,  we  can  obtain  the  planet's  heliocentric 
longitude,  as  in  Art.  182.  This  is,  as  we  have  already  seen, 
the  heliocentric  longitude  of  the  node. 

The  nodes  of  every  planet  are  found  to  have  a  westward 
movement,  similar  in  character  to  the  precession  of  the 
equinoxes.  It  is  a  very  slight  movement,  however,  being 
only  70'  a  century  in  the  case  of  Mercury,  and  being  less  than 
that  in  the  case  of  the  other  planets. 

191.  Inclination  of  the  Planet's  Orbit  to  the  Ecliptic. 
— The  line  of  nodes  of  any  planet  being,  as  we  have  just  now 
seen,  a  nearly  stationary  line  in  the 
plane  of  the  ecliptic,  the  earth  must 
pass  it  once  in  very  nearly  six 
months  in  its  revolution  about  the 
sun.  The  inclination  of  the  plane  of  i 
any  planet's  orbit  to  the  plane  of 
the  ecliptic  may  be  determined  by 
observations  made  when  the  earth  is 
on  the  line  of  nodes.  In  Fig.  67  let  E  be  the  earth,  and 
EN  the  line  of  nodes  of  a  planet.  Let  A  EN  be  the  plane 
of  the  ecliptic,  and  P  the  position  of  a  planet  projected  on 
the  surface  of  the  celestial  sphere.  With  EP  as  a  radius, 
let  the  arc  PA  be  described,  perpendicular  to  the  plane 
of  the  ecliptic,  and  also  the  arcs  PN  and  NA>  In  the 
spherical  triangle  PNA,  right-angled  at  A,  the  arc  PA, 
which  measures  the  angle  PEA,  is  the  geocentric  latitude 
of  the  planet;  AN}  or  the  angle  AEN,  is  the  difference 
between  the  geocentric  longitudes  of  the  planet  and  the 


180  THE  PLANETS  AND  THE  PLANETOIDS 

node;  and  the  angle  PNA  is  the  inclination  of  the  plane 
of  the  orbit  to  the  plane  of  the  ecliptic.  In  the  triangle 
PNA,  we  have, 

tan  PA 

sin  AN' 

It  may  be  noticed  in  this  formula  that,  since  the  sine  of  a 
small  angle  varies  more  rapidly  than  the  sine  of  a  large  angle, 
an  error  in  A  N  will  affect  the  result  the  less,  the  greater  A  N 
itself  happens  to  be  at  the  time  of  observation:  that  is  to  say, 
the  farther  the  planet  is  from  the  node. 

192.  The  Periodic  Time  of  an  Inferior  Planet.— The  time 
in  which  an  inferior  planet  makes  one  complete  revolution 
about  the  sun,  or  its  periodic  time,  may  be  found  by  taking  the 
interval  of  time  between  two  successive  passages  of  the 
planet  through  the  same  node.  The  accuracy  of  this  method 
is,  however,  diminished  by  the  small  inclination  (less  than 
7°  of  the  planes  of  the  orbits  to  the  plane  of  the  ecliptic, 
which  renders  it  difficult  to  determine  the  instant  when  the 
latitude  of  the  planet  is  zero.  It  is  found  to  be  better  to 
determine  the  planet's  synodical  period,  or  the  interval 
between  two  successive  conjunctions  of  the  same  kind,  and 
from  it  to  compute  the  sidereal  period.  The  conditions  of 
this  problem,  and  the  method  by  which  it  is  solved,  are 
identical  with  those  in  the  case  of  the  synodical  and  the 
sidereal  period  of  the  moon,  Art.  141.  The  formula  there 
given  was, 

ST 


P= 


s+r 


In  applying  this  to  the  case  of  an  inferior  planet,  P  and  S 
denote  the  sidereal  and  the  synodical  period  of  the  planet, 
and  T  denotes,  as  before,  the  sidereal  year. 

Instead  of  using  the  interval  between  two  conjunctions 
as  the  synodical  period,  we  may  take  the  interval  between 
two  greatest  elongations  of  the  same  kind.  A  very  accurate 


MERCURY  181 

mean  synodical  period  is  obtained  by  taking  two  elongations 
separated  by  a  long  interval,  and  dividing  this  interval  by 
the  number  of  synodical  periods  which  it  contains 

193.  It  is  hardly  necessary  to  describe  the  methods  by 
which  the  other  elements  given  in  Art.  189  are  determined. 
It  will  readily  be  understood  that  the  distance  of  the  planet 
from  the  sun.  obtained  as  in  Art.  190,  or  by  Kepler's  Law, 
will  enable  us  to  obtain  the  form  and  the  magnitude  of  the 
ellipse  in  which  the  planet  moves.     The  method  by  which 
the  longitude  of  the  perihelion  is  obtained,  although  not 
intrinsically  difficult,  is  too  elaborate  for  this  work.     Finally, 
when  the  longitude  of  the  perihelion  is  obtained,  the  time  of 
the  planet's  perihelion  passage  may  evidently  at  once  be 
determined      The  perihelion  of  Venus  has  a  very  minute 
retrograde  motion :  the  perihelia  of  all  the  other  planets  have 
an  eastward  motion,  similar  to  that  of  the  earth's  line  of 
apsides. 

MERCURY 

194.  Mercury  revolves  about  the  sun  at  a  mean  distance 
of  about  36,000,000  miles,  the  eccentricity  of  its  orbit  being 
about  ^th      Its  synodical  period  is  about  116  days,  and  its 
sidereal  period  88  days.     Its  real  diameter  is  about  3000 
miles,*     Its  mass  is  a  matter  of  considerable  uncertainty, 
and  quite  different  values  of  it  are  given  by  different  astron- 
omers.    It  may  be  assumed  to  be  approximately  equal  to 
aV  th  of  the  mass  of  the  earth. 

The  greatest  elongation  of  Mercury  is  only  about  28°  20': 
and  hence  the  planet  is  rarely  visible  to  the  naked  eye, 

*  The  distances  and  the  diameters  of  all  the  celestial  bodies  except 
the  moon  depend  for  their  accuracy  upon  the  accuracy  with  which 
the  solar  parallax  is  determined.  An  error  of  1"  in  this  parallax  would 
affect  the  sun's  distance  from  us  to  the  amount  of  several  millions  of 
miles  and  proportionally  also  the  distances  and  the  diameters  of  the 
other  celestial  bodies;  the  values  given  must  therefore  be  considered 
to  be  only  approximate. 


182  THE  PLANETS  AND  THE  PLANETOIDS 

and  is  never  a  conspicuous  object  in  the  heavens.  Its 
spectrum  shows  lines  which  seem  to  imply  an  atmosphere 
containing  water  vapor.  It  exhibits  phases  similar  to 
those  of  the  moon,  and  due  to  the  same  causes.  It  was 
asserted  by  some  observers  that  it  rotates  on  an  axis  in  about 
24  hours,  but  later  observations  fail  to  confirm  this  and 
modern  astronomers  are  still  unable  to  determine  the  period 
of  rotation  of  the  planet.  If  it  has  any  compression,  it  is 
extremely  small. 

VENUS 

195.  The  mean  distance  of  Venus  from  the  sun  is  about 
67,000,000  miles,  its  synodical  period  is  584  days,  and  its 
sidereal  period  225  days.     The  eccentricity  of  its  orbit  is 
small,  being  only  about  x^Voths.     Venus  is  nearly  as  large 
as  the  earth,  its  diameter  being  7700  miles.     Its  mass  is 
about  f  ths  of  the  mass  of  the  earth.     It  has  no  perceptible 
compression. 

The  greatest  elongation  of  Venus  from  the  sun  amounts 
to  about  47°  15',  and  hence  it  is  often  visible  as  an  evening  or 
a  morning  star.  At  certain  times  its  brightness  is  so  great 
that  it  can  be  seen  in  broad  daylight  with  the  naked  eye, 
while  at  night  shadows  are  cast  by  the  objects  which  it 
illuminates.  It  exhibits  phases  similar  to  those  of  Mercury. 

It  seems  to  be  generally  admitted  that  Venus  has  an 
atmosphere  the  density  of  which  is  not  very  different  from 
that  of  the  earth's  atmosphere.  Observations  of  spots  upon 
the  disc  go  to  show  that  the  planet  rotates  upon  an  axis  in  a 
period  of  about  23 J  hours;  but  this  conclusion  is  by  no  means 
certain. 

The  existence  of  a  satellite  of  Venus  was  formerly  sus- 
pected, but  no  satellite  was  seen  at  the  transits  in  1874  and 
1882. 

196.  Transits  of  Venus.— We  have  already  seen  (Art.  93) 
how  a  transit  of  Venus  across  the  sun's  disc  is  employed  in 


VENUS  183 

determining  the  distance  of  the  earth  from  the  sun.  If  the 
plane  of  the  orbit  of  Venus  coincided  with  the  plane  of  the 
ecliptic,  a  transit  would  occur  whenever  the  planet  came  into 
inferior  conjunction,  or  once  in  every  584  days.  Owing  to 
the  inclination  of  the  planes  to  each  other,  however,  it  is 
evident  that  at  the  time  of  inferior  conjunction  the  planet 
may  have  too  great  a  latitude  to  touch  any  part  of  the  sun's 
disc.  Now  the  phenomenon  of  a  transit  of  Venus  is  analogous 
to  a  solar  eclipse,  and  therefore  if  in  Fig.  60  we  suppose  M 
to  be  Venus,  the  formula  obtained  in  Art.  160  will  apply 
equally  well  to  the  limits  of  the  geocentric  latitude  of  Venus 
within  which  a  transit  is  possible.  These  limits  are  the  sum 
of  the  semi-diameters  of  the  sun  and  the  planet  and  of  the 
parallax  of  the  planet,  diminished  by  the  parallax  of  the 
sun.  The  greatest  value  of  the  limits  will  be  found  to  be 
about  11'  49".  When,  therefore,  the  latitude  of  Venus  is 
more  than  17'  49"  at  the  time  of  inferior  conjunction,  no 
transit  will  occur:  and  as  Venus  in  every  sidereal  revolution 
attains  a  latitude  of  over  3°  23',  it  is  at  once  evident  that  a 
transit  is  only  a  rare  occurrence. 

197.  Intervals  between  Transits.  —  Since  the  latitude  of 
Venus  is  so  small  when  a  transit  occurs,  it  is  plain  that  the 
planet  must  be  either  at  or  very  near  one  of  its  nodes.  Now, 
let  us  suppose  that  Venus  is  at  its  node  at  the  time  of  inferior 
conjunction,  under  which  circumstances  a  transit  will,  of 
course,  take  place.  The  sidereal  period  of  Venus  is  224.7 
days.  Now,  we  have, 


224.7d.     Xl3  =  2921.11d.;  and, 
365.256d.X  8  =  2922.05d. 

At  the  end  of  eight  years,  then,  Venus  will  be  very  near  the 
same  node  at  the  time  of  inferior  conjunction,  and  a  transit 
will  probably  occur.  Again,  we  have. 

224.7d.     X382  =  85835.4d. 
365.256d.  X  235  =  85835.  16d.  ; 


184  THE  PLANETS  AND  THE  PLANETOIDS 

so  that  transits  at  tne  same  node  also  occur  every  235  years. 
In  the  same  way  transits  may  also  occur  at  the  other  node : 
and  the  intervals  between  transits  at  either  node  are  found 
to  be  8,  105J,  8,  121J,  8,  etc.,  years.*  The  longitude  of  the 
ascending  node  of  Venus  is  75°  20',  and  a  transit  at  that  node 
must  occur,  when  it  occurs  at  all,  at  the  time  when  the  sun 
is  near  that  point,  which  is  about  the  6th  of  June.  For  a 
similar  reason  transits  at  the  descending  node  occur  about 
the  6th  of  December. 

The  last  three  transits  occurred  in  June,  1769,  December, 
1874,  and  December,  1882.  The  next  transits  will  occur  in 
June,  2004,  and  June,  2012.  (See  Table  VIII.,  Appendix.) 

SUPERIOR  PLANETS 

198.  The  superior  planets  are,  as  they  have  already  been 
defined  to  be,  planets  whose  orbits  include  the  orbit  of  the 
earth.     They  have,  like  the  inferior  planets,  superior  con- 
junction, but  can  evidently  have  no  inferior  conjunction. 
Their  elongation  from  the  sun,  eastern  and  western,  can  have 
all  values  between  0°  and  180°.     When  their  elongation  is 
90°,  they  are  said  to  be  in  quadrature,  and  when  it  is  180°,  in 
opposition.     Much  that  has  already  been  said  in  this  chapter 
in  reference  to  the  inferior  planets  is  equally  true  in  reference 
to  the  superior  planets.     The  elements  of  the  orbits  are  in 
both  instances  the  same,  and  so  are  the  methods  by  which 
the  heliocentric  longitudes  of  the  nodes  and  the  inclinations 
of  the  orbits  are  determined.     In  some  other  points  there  is  a 
difference  between  the  two  classes  of  planets;  and  these 
points  we  shall  now  proceed  to  examine. 

199.  Retrograde  Motion. — The  superior  planets,  like  the 
inferior  planets,  have  at  times  an  apparent  retrograde  motion, 
which  occurs  at  or  near  the  time  of  opposition.     The  explana- 
tion of  this  retrogradation  will  be  seen  by  a  reference  to  Fig. 

*  Thus  the  years  of  transits  at  the  ascending  node  are  1761,  1769, 
and  2004:  at  the  descending  node,  1639,  1874,  and  1882. 


SUPERIOR  PLANETS 


185 


68.  S  is  the  sun,  EE'E"E"'  the  orbit  of  the  earth,  and 
CGDP  the  orbit  of  a  superior  planet,  the  plane  of  which  is 
supposed  to  coincide  with  the  plane  of  the  ecliptic,  and  to 
meet  the  celestial  sphere  in  the  circle  ANBM.  Let  the 
earth  be  at  E,  and  the  planet  at  P,  180°  in  geocentric  longi- 
tude from  the  sun.  The  planet  will  appear  to  be  projected 
upon  the  celestial  sphere  at  the  point  M.  Let  both  earth  and 
planet  revolve  in  their  orbits  in  the  direction  indicated  by 


M'MM"M' 
FIG.  68. 

the  arrow.  When  the  earth  has  reached  the  point  E',  the 
planet,  whose  angular  velocity  is  less  than  that  of  the  earth, 
will  have  reached  some  point  P',  and  will  lie  in  the  direction 
E'M' :  in  other  words,  it  has  apparently  retrograded.  If,  on 
the  other  hand,  the  earth  is  at  E"  and  the  planet  at  P,  or  in 
superior  conjunction,  the  apparent  motion  of  the  planet  is 
at  that  point  direct,  and  its  angular  velocity  appears  to  be 
greater  than  it  really  is;  for  when  the  earth  is  at  E'",  and  the 
planet  at  P',  the  latter,  having  in  reality  moved  through  the 


186  THE  PLANETS  AND  THE  PLANETOIDS 

arc  MM"  since  conjunction,  appears  to  have  moved  through 
the  arc  MM'". 

The  apparent  motion,  then,  of  a  superior  planet  is  direct 
in  all  cases  except  when  it  is  at  or  near  its  opposition.  The 
apparent  motion  of  an  inferior  planet  has  been  shown  to  be 
retrograde  at  and  near  the  time  of  inferior  conjunction 
(Art.  186).  Now,  since  the  earth  is  a  superior  planet  to  an 
inferior  planet,  and  an  inferior  planet  to  a  superior  planet,  we 
see,  by  comparing  the  two  cases,  that  the  retrograde  motion 
occurs  in  each  class  of  planets  at  and  near  the  time  when  the 
inferior  planet  comes  between  the  sun  and  the  superior  planet. 

The  stationary  points  in  the  orbit  of  a  superior  planet 
are  identical  in  character  with  those  in  the  orbit  of  an  inferior 
planet,  and  occur  when  the  retrograde  motion  is  changing  to 
the  direct  motion,  or  the  direct  to  the  retrograde. 

200.  Synodical  and  Sidereal  Periods. — The  synodical 
period  of  a  superior  planet  is  the  interval  of  time  between 
two  successive  conjunctions  or  two  successive  oppositions. 
When  a  planet  is  in  conjunction  with  the  sun,  it  is  above  the 
horizon  only  in  the  day  time;  but  when  it  is  in  opposition,  it  is 
above  the  horizon  during  the  night,  and  can  therefore  be 
readily  observed.  Hence  in  obtaining  the  synodical  period 
it  is  better  to  employ  the  interval  of  time  between  two 
oppositions;  and  by  determining  the  times  of  two  oppositions 
which  are  not  consecutive,  and  dividing  the  interval  between 
them  by  the  number  of  synodical  revolutions  which  it  con- 
tains, we  may  obtain  a  mean  value  of  the  synodical  period. 
By  using  times  of  opposition  which  were  observed  and 
recorded  before  the  Christian  era,  a  very  accurate  value  of 
the  mean  synodical  period  may  be  obtained. 

The  method  of  deducing  the  periodic  time  from  the 
synodical  period  is  the  same  that  was  used  in  the  case  of  the 
inferior  planets  (Art.  192),  with  the  important  exception 
that  in  the  present  case  it  is  the  earth  that  gains  360°  upon 
the  planet  in  the  course  of  a  synodical  revolution,  and  not 
the  planet  that  gains  it  upon  the  earth.  If,  therefore,  we 


SUPERIOR  PLANETS  187 

denote  the  periodic  times  of  the  earth  and  the  planet  by  T 
and  P,  and  the  synodical  period  of  the  planet  by  S,  we  shall 
have  (Art.  141), 

360°    360°    360° 
T  "   P   '     S 

ST 

s-r 

which  gives  the  value  of  any  planet's  sidereal  period  in  terms 
of  its  synodical  period  and  the  sidereal  year. 

201.  Distance  of  a  Superior  Planet  from  the  Sun. — The 
distance  of  a  superior  planet  from  the  sun  may  be  obtained 
by  the  method  of  Art.  190: 
or  it  may  be  obtained  from 
observations    made  at  the 
time    it   is    in    opposition. 
In  Fig.  69,  let  S  be    the     ^~  ~~E p" 

sun,    E  the   earth,  and  P  FIG.  69. 

a    superior    planet   at   the 

time  of  opposition,  the  planes  of  the  two  orbits  being  supposed 
to  coincide.  At  the  end  of  a  short  interval,  let  the  earth 
have  moved  to  Ef,  and  the  planet  to  P';  the  angle  E'OE  will 
be  the  amount  of  the  apparent  retrogradation  of  the  planet 
in  that  time.  The  periods  of  the  earth  and  the  planet  being 
known,  we  can  compute  the  angular  advance  of  each  planet 
in  the  given  interval,  thus  obtaining  the  angles  E'SE  and 
P'SP.  The  radius  vector  of  the  earth's  orbit,  SE'  can  be 
found  from  the  Ephemeris.  Then,  in  the  triangle  E'SP', 
we  know  the  side  E'S,  the  angle  E ,'SP' ',  which  is  the  angular 
gain  of  the  earth  on  the  planet,  and  the  angle  E'P'S,  which 
is  the  sum  of  P'SO,  or  the  advance  of  the  planet,  and  P'OS, 
or  the  apparent  retrogradation  of  the  planet.  We  can 
therefore  compute  the  side  SP*,  which  is  the  distance 
required. 

If  we  suppose  the  real  distance  of  the  sun  from  the  earth 
not  to  be  known,  this  method  will  still  give  us  the  ratio  • 


188  THE  PLANETS  AND  THE  PLANETOIDS 

between  this  distance  and  the  distance  of  the  planet  from  the 
sun.  And  furthermore,  if  we  can  determine  the  real  dis- 
tance of  the  planet  from  the  earth  by  observations  of  its 
parallax  at  the  time  of  opposition  (when  it  is  nearest  to  the 
earth),  by  obtaining  its  displacement  in  right  ascension  when 
far  east  and  far  west  of  the  same  meridian,  we  are  able  to 
obtain  the  real  distance  of  the  earth  from  the  sun.  Such 
observations  have  been  made  upon  the  planet  Mars,  and  the 
distance  of  the  earth  from  the  sun  has  been  deduced. 

202.  Evening  and  Morning  Stars. — The  angular  velocity 
of  a  superior  planet  towards  the  east  is  less  than  that  of  the 
earth,  and  consequently  also  less  than  the  sun's  apparent 
angular  velocity  in  the  same  direction.     After  conjunction, 
therefore,  the  planet  will  lie  to  the  west  of  the  sun,  and  its 
elongation  will  continually  increase.     When  this  elongation 
exceeds  about  30°,  the  planet  will  begin  to  be  visible  as  a 
morning  star,  and  will  so  continue  until  it  has  fallen  180° 
to  the  west  of  the  sun,  and  is  in  opposition.     It  will  then  rise 
about  sunset   and  set  about  sunrise.     After  the  time  of 
opposition  it  will  lie  more  than  180°  to  the  west  of  the  sun,  or, 
what  is  the  same  thing,  less  than  180°  to  the  east  of  it,  and 
will  rise  before  sunset.     It  will  therefore  be  an  evening  star 
from  opposition  to  conjunction. 

MARS 

203.  The  synodical  period  of  Mars  is  780  days,  and  its 
sidereal  period  687  days.     Its  mean  distance  from  the  sun  i>« 
141,551,000  miles,  and  the  eccentricity  of  its  orbit  about 
Ti-th.     Its  diameter  is  4549  miles.     Different  values    are 
given  to  its   compression,   the  latest   observations    giving 
about  2To"th.     Its  mass  is  about  ij-th  of  that  of  the  earth.     It 
has  two  very  small  satellites,  discovered  by  Prof.  A.  Hall, 
U.  S.  Navy,  in  1877. 

204.  Phases. — At  opposition  and  conjunction  the  same 
Hemisphere  is  turned  towards  both  the  sun  and  the  earthy 


MARS  189 

and  consequently  the  planet  appears  full.  At  quadrature  it 
appears  slightly  gibbous.  It  is  the  only  one  of  the  superior 
planets  which  exhibits  any  sensible  phases,  excepting  pos- 
sibly Jupiter. 

Mars  shines  with  a  red  light,  and  at  opposition  is  a  veiy 
conspicuous  object,  sometimes  equaling  Jupiter  in  bril- 
liancy. 

205.  Rotation,  etc. — When  examined  in  a  telescope,  the 
surface  of  Mars  is  seen  to  be  covered  with  patches  of  a  dull 
reddish  color,  interspersed  with  spots  of  a  bluish  or  greenish 
hue.     By  observation  of  these  spots  Mars  is  found  to  rotate 
upon  an  axis  once  in  about  24^  hours.     The  axis  of  rotation 
is  inclined  at  an  angle  of  65°  to  the  plane  of  the  planet's 
orbit,  and  hence  there  must  be  a  change  of  seasons  not  very 
different  from  the  change  which  takes  place  on  the  earth. 
White  spots  are  seen  near  the  poles,  which  decrease  in  the 
Martian  summer  and  increase  in  the  winter.    These  spots  are 
supposed  to  be  ice,  but  there  is  quite  a  possibility  that  these 
caps  are  not  ice  at  all  but  some  other  substance  such  as 
solidified  carbon  dioxide.     The  probability  that  Mars  has 
inhabitants  more  or  less  like  those  of  the  earth  hinges  largely 
on  the  identity  of  physical  conditions,  atmosphere,  tempera- 
ture, moisture,  etc.,  in  the  two  planets.     From  observations 
made  in   1909  it  was  proved  that  whatever  atmosphere 
exists  on  Mars  contains  almost  no  water  -vapor  and  most 
astronomers  are  now  agreed  that  there  is  little  or  no  atmos- 
phere on  the  planet.     The  theoretic  temperature  of  Mars 
is  —33°  Fahrenheit  and   this  would  negative  the  idea  of 
water  except  in  the  form  of  ice, 

THE  MINOR  PLANETS 

206.  Bode's  Law. — In    1778    the    astronomer    Bode    of 
Berlin  announced  (though  he  did  not  discover)  the  following 
curious  relation  between  the  distances  of  the  different  planets 
from  the  sun.     The  statement  of  this  relation  usually  goes 


190  THE  PLANETS  AND  THE  PLANETOIDS 

by  the  name  of  "Bode's  Law."  If  we  take  the  series  of 
numbers 

0,  3,  6,  12,  24,  48,  96,  192,  384, 

each  of  which,  except  the  second,  is  double  the  preceding 
one,  and  add  4  to  each  of  these  numbers,  the  resulting  series, 

4,  7,  10,  16,  28,  52,  100,  196,  388, 

will  approximately  represent  the  relative  distances  of  the 
planets  from  the  sun.*  Thus  Mercury  is  36,000,000  miles 
from  the  sun,  and  Venus  67,000,000  miles;  and  these  distances 
are  to  each  other  nearly  in  the  ratio  of  4  to  7.  There  was, 
however  (the  minor  planets  being  then  undiscovered),  a 
break  in  the  series,  there  being  no  planet  corresponding  to 
the  number  28;  and  Bode  ventured  to  predict  that  another 
planet  might  be  found  to  exist  at  that  point  of  the  series: 
that  is  to  say,  between  Mars  and  Jupiter.  A  similar  pre- 
diction was  made  by  Kepler,  about  the  beginning  of  the 
seventeenth  century. 

207.  Discovery  of  the  Minor  Planets.— In  1800  six 
European  astronomers  formed  an  association  for  the  express 
purpose  of  searching  the  heavens  for  new  planets;  and  within 
the  next  six  years  four  minor  planets  were  discovered.  These 
were  named  Ceres,  Pallas,  Juno,  and  Vesta.  No  more  were 
discovered  until  the  end  of  1845,  but  since  that  time  some 
have  been  discovered  in  nfearly  every  year.  The  number 
discovered  up  to  January  1,  1917  (including  Ceres,  etc.),  was 
more  than  800. 

The  mean  distances  of  these  bodies  from  the  sun  vary  from 
198,000,000  to  400,000,000  miles.  They  are  all  very  small, 
the  largest  being  probably  not  over  300  miles  in  diameter, 
and  many  of  the  others  being  too  small  to  admit  of  measure- 

*  The  last  two  numbers  were  not  in  the  series  as  originally  announced 
by  Bode,  since  Uranus  and  Neptune  had  not  then  been  discovered. 
The  real  distance  of  Neptune  is  one-fourth  less  than  it  should  be,  if 
this  law  were  anything  more  than  a  coincidence. 


THE  MINOR  PLANETS  191 

ment.  Vesta  is  the  only  one  which  is  ever  visible  to  the 
naked  eye,  and  its  visibility  is  very  rare.  Some  of  the  others 
are  so  small  that  they  can  scarcely  be  seen  with  the  strongest 
telescope,  even  at  opposition.  Their  total  mass  is  about 
one-third  of  the  earth's. 

The  French  astronomer  Leverrier  has  concluded  that  the 
mass  of  these  minor  planets  is  by  no  means  sufficient  to 
produce  the  perturbations  in  the  orbit  of  Mars  and  in  that  of 
Jupiter  which  are  believed  to  be  due  to  the  attractions  of 
this  group.  It  is  therefore  extremely  probable  that  many 
other  planets,  hitherto  undiscovered,  belong  to  the  same 
cluster. 

208.  Olbers's  Theory,— Shortly  after  the  discovery  of 
the  first  four  minor  planets,  Dr.  Olbers  advanced  the  theory 
that  these  planets  were  fragments  of  a  single  planet,  which 
had  been  broken  in  pieces  by  volcanic  action  or  by  some 
other  internal  force.     This  theory  requires  that  the  orbits 
should  have  now,  or  should  have  had  at  some  former  period, 
a  common  point  of  intersection;  but  no  such  common  point 
has  been  found.     "So  far  as  can  be  judged,  these  bodies 
have  been  revolving  about  the  sun  as  separate  planets  ever 
since  the  solar  system  itself  was  formed."     (Newcomb.) 

JUPITER 

209.  Jupiter  is  the  largest  planet  of  our  system.     At 
times  it  surpasses  Venus  in  brilliancy,  and  even  casts  a 
shadow.     It  revolves  about  the  sun  at  a  mean  distance  of 
483,853,000  miles.     The  eccentricity  of  its  orbit  is  about 
•aVth.     Its  synodical  period  is  399  days,  and  its  sidereal 
period   4333    days,    or   about    11.9   years.     Its    equatorial 
diameter  is  about  90,000  miles,  and  its  volume  is  about  1390 
times  that  of  the  earth.     It  rotates  on  an  axis  in  a  little  less 
than  ten  hours,  and  has  a  compression  of  r^th.     Its  phases 
are  so  slight  as  to  be  scarcely  perceptible. 

210.  Belts. — When  examined  through  a  telescope,   the 


192  THE  PLANETS  AND  THE  PLANETOIDS 

disc  of  Jupiter  is  seen  to  be  streaked  with  dark  belts,  lying 
nearly  parallel  to  the  plane  of  its  equator.  With  powerful 
telescopes  these  belts  are  found  to  have  a  gray  or  brown  tinge. 
They  are  sometimes  nearly  permanent  for  several  months, 
and  sometimes  they  change  their  shape  materially  in  the 
course  of  a  few  minutes.  There  are  usually  one  broad  and 
several  narrower  belts  on  each  side  of  Jupiter's  equator. 

It  is  generally  supposed  that  Jupiter  is  surrounded  by  a 
dense  atmosphere,  and  that  these  belts  are  fissures  in  this 
atmosphere,  through  which  the  dark  body  of  the  planet  is 
seen.  The  distribution  of  the  atmosphere  in  lines  so  nearly 
parallel  to  the  equator  is  supposed  to  be  due  to  currents  in 
the  atmosphere,  similar  in  character  to  our  trade-winds,  but 
having  a  more  decided  easterly  and  westerly  tendency,  from 
the  more  rapid  rotation  and  the  greater  size  of  the  planet. 
A  point  on  Jupiter's  equator  rotates  with  a  velocity  of  about 
28,000  miles  an  hour,  while  a  point  on  our  own  equator 
rotates  with  a  velocity  of  only  about  24,000  miles  a  day. 

The  rapidity  of  changes  upon  the  visible  surface  implies 
the  expenditure  of  considerably  more  heat  than  the  planet 
could  possibly  receive  from  the  sun  and  astronomers  have 
come  to  the  conclusion  that  Jupiter  is  at  a  temperature  very 
close  to  incandescence  if  it  is  not  a  white-hot  liquid  or  gaseous 
globe.  It  seems  probable  that  very  little,  if  any,  light  is 
given  off  by  the  planet  itself  as  the  satellites  •  disappear 
completely  when  they  enter  the  shadow. 

211.  Satellites. — Jupiter  is  attended  by  nine  satellites 
or  moons,  seven  revolving  about  it  from  west  to  east,  and 
two  from  east  to  west.  Of  the  first  four  discovered,  dis- 
tinguished from  each  other  by  the  numbers,  I.,  II.,  III.,( 
and  IV.,  the  first  satellite  is  the  nearest  to  Jupiter  of  those 
found  at  this  time.  The  second  satellite  is  about  as  large 
as  our  moon,  and  the  others  are  somewhat  larger.  They 
are  not  usually  visible  to  the  naked  eye,  though  a  few  in- 
stances to  the  contrary  are  on  record.  The  distance  of  the 
first  satellite  from  Jupiter  is  260,000  miles,  and  that  of  the 


JUPITER  193 

fourth  is  1,162,000  miles.  /The  first  revolves  about  Jupiter  in 
a  period  of  42  hours,  and  the  fourth  in  a  period  of  16d.  18h. 

The  first  four  satellites  were  discovered  by  Galileo,  in 
1610.  The  fifth  was  discovered  in  1892,  by  Mr.  E.  E. 
Barnard  of  the  Lick  Observatory.  It  seems  to  be  not  more 
than  100  miles  in  diameter  and  its  mean  distance  from  the 
surface  of  Jupiter  is  only  about  68,000  miles.  The  sixth, 
seventh,  eighth,  and  ninth  satellites  are  very  small. 

212.  Phenomena  Presented  by  the  Satellites.— The 
satellites,  in  the  course  of  their  revolution  about  their  pri- 
mary, present  four  distinct  classes  of  phenomena,  which  are 


FIG.  70. 

shown  in  Fig.  70.  In  this  figure  let  S  be  the  disc  of  the  sun, 
and  EE'E"E"'  the  orbit  of  the  earth.  Let  J  be  Jupiter, 
and  ABDG  the  orbit  of  one  of  its  satellites.  Since  the 
planes  of  all  the  orbits  very  nearly  coincide  with  the  plane 
of  the  ecliptic,  we  may  consider  ABDG  to  lie  in  that  plane. 
Suppose  the  earth  to  be  at  E,  and,  in  order  to  simplify  the 
case,  suppose  it  also  to  remain  at  that  point  during  the  short 
time  required  by  the  satellite  to  revolve  about  Jupiter. 
An  eclipse  of  the  satellite  will  occur  when  it  passes  through 
the  arc  M  N,  since  it  is  then  within  the  shadow  formed  by 
lines  drawn  tangent  to  the  disc  of  the  sun  and  that  of  Jupiter. 
It  may  readily  be  calculated  that  the  length  of  the  shadow, 
from  J  to  C,  is  about  55,000,000  miles,  so  that  the  shadow 
extends  far  beyond  the  orbit  of  the  fourth  satellite.  In 


194  THE  PLANETS  AND  THE  PLANETOIDS 

extremely  rare  cases  this  satellite,  owing  to  the  inclination 
of  its  orbit  to  the  ecliptic,  may  fail  to  be  eclipsed. 

An  occultation  of  the  satellite  will  occur  when  it  passes 
through  the  arc  AB,  since  it  is  then  within  the  cone  formed 
by  lines  drawn  from  E  tangent  to  the  disc  of  Jupiter. 

A  transit  of  the  shadow  will  occur  when  the  satellite  passes 
through  the  arc  GH,  its  shadow  being  then  cast  upon  the 
disc  of  Jupiter,  and  moving  across  it  as  a  small  round  spot. 

Finally,  a  transit  of  the  satellite  will  occur  when  it  passes 
through  the  arc  KL. 

It  will  evidently  depend  on  the  relative  situation  of  the 
sun,  the  earth,  and  Jupiter,  whether  all  these  phenomena 
will  be  observed  or  not.  When  the  earth,  for  instance,  is 
at  E'  or  E'" ,  it  is  plain  that  only  an  occultation  and  a  transit 
will  occur. 

The  relative  situation  of  the  satellites  to  each  other  and  to 
their  primary  are  constantly  changing.  Sometimes  all  are 
on  the  same  side  of  the  primary;  sometimes  only  one  is 
visible,  and  sometimes,  though  very  rarely,  all  are  invisible. 
The  times  at  which  these  different  phenomena  will  occur 
are  computed  beforehand,  and  are  given  in  the  American 
Ephemeris,  the  time  used  being  that  of  the  meridian  of 
Washington.  The  longitude  of  any  place  can  therefore  be 
obtained,  at  least  approximately,  by  observations  of  these 
phenomena. 

213.  Velocity  of  Light. — If  the  transmission  of  light  is  not 
instantaneous,  it  is  evident  that  the  same  phenomenon,  if 
observed  both  at  E'  and  E'"  (Fig.  70),  will  not  occur  at  the 
same  absolute  instant  of  time  at  both  places,  but  will  occur 
later  at  E"'  by  the  time  required  for  light  to  cross  the  orbit 
of  the  earth,  a  distance  of  185,000,000  miles.  And  such  is 
actually  the  case.  This  peculiarity  was  first  noticed  by 
Romer,  a  Danish  astronomer,  in  1675,  who  found  that  the 
times  at  which  the  phenomena  occurred  were  earlier  by 
about  eight  minutes  at  E',  and  later  by  the  same  amount  at 
E'" ,  than  the  times  computed  for  the  mean  distance  of 


JUPITER  195 

Jupiter  from  the  sun.  The  time  required  by  light  in  passing 
from  E'  to  E"r  has  been  found  by  observation  to  be  very 
nearly  16m.  27s.;  whence  the  velocity  of  light  is  calculated 
to  be  187,000  miles  a  second,  a  result  agreeing  very  closely 
with  the  velocity  obtained  from  the  constant  of  aberration, 
discussed  in  Chapter  VIII. 

214.  Mass  of  Jupiter. — The  mass  of  Jupiter  is  much 
more  accurately  known  than  the  mass  of  any  of  the  planets 
which  have  hitherto  been  described.  The  reason  is  that 
Jupiter  is  attended  by  satellites  whose  distances  from  their 
primary,  and  whose  periods  of  revolution,  can  be  obtained  by 
observation.  We  are  thus  enabled  to  compare  directly  the 
attraction  which  Jupiter  exerts  on  one  of  its  satellites  with 
the  attraction  which  the  sun  exerts  on  Jupiter;  and  as,  by 
the  law  of  gravitation,  the  ratio  of  these  two  attractions  is 
directly  as  the  ratio  of  the  masses  of  the  two  attracting 
bodies,  and  inversely  as  the  square  of  the  ratio  of  the  distances 
through  which  these  attractions  are  exerted,  it  is  evidently 
within  our  power  to  obtain  the  ratio  of  the  two  masses. 
Since  the  attraction  of  the  sun  on  Jupiter  is  equal  to  the 
centrifugal  force  of  Jupiter  in  its  orbit,  if  we  denote  the  dis- 
tance of  Jupiter  from  the  sun  by  D  and  its  sidereal  period 
by  T7,  we  have,  by  the  formula  of  Art.  69,  the  expression  for 

the  sun's  attraction  on  Jupiter  equal  to          .     In  the  same 

way,  denoting  the  distance  of  a  satellite  from  Jupiter  by  d, 
and  its  period  by  t,  we  have  for  the  attraction  of  Jupiter  on 

A      9  ,7 

the  satellite,  —&-,  so  that  the  ratio  of  the  two  attractions 

Dt2 
is  — 2-     Finally,  denoting  the  mass  of  the  sun  by  M ,  and 

that  of  Jupiter  by  m,  we  shall  have, 
DP^M  d? 
dT2  mXD2' 

M     D*t2 
"    m    d3T2' 


196  THE  PLANETS  AND  THE  PLANETOIDS 

By  this  formula  an  approximate  value  can  be  obtained  of  the 
mass  of  any  planet  which  is  attended  by  a  satellite. 

The  mass  of  Jupiter  is  found  to  be  r^sth  of  that  of  the 
sun,  or  about  316  times  that  of  the  earth. 

SATURN 

215.  Saturn  is,  next  to  Jupiter,  the  largest  planet  of  our 
system,  and  may  fairly  be  considered  to  be  the  most  interest- 
ing.    It  revolves  about  the  sun  at  a  mean  distance  of  887,- 
000,000  miles,  in  an  orbit  whose  eccentricity  is  about  xVth. 
Its  synodical  period  is  378  days  and  its  sidereal  period  29.46 
years.     Its  diameter  is  76,456  miles,  and  it  has  a  compression 
of  about  Jth.     It  is  attended  by  ten  satellites,  the  planes  of 
whose  orbits,  with  one  exception,  very  nearly  coincide  with 
the  plane  of  its  equator.     Four  of  these  satellites  are  rarely 
visible  in  any  but  the  strongest  telescopes.     Their  distances 
from  the  planet  range  from  121,000  miles  to  8,000,000  miles, 
and  their  sidereal  periods  from- 22  hours  to  over  523  days. 
All  but  one  appear  to  be  smaller  than  our  moon. 

216.  Rotation,  etc. — Saturn  rotates  upon  an  axis  which 
is  inclined  at  an  angle  of  about  63°  to  the  plane  of  its  orbit, 
in  a  period  of  10J  hours.     Belts  are  seen  on  the  body  of  the 
planet,  similar  to  those  of  Jupiter,  although  less  marked. 
Other  indications  of  the  existence  of  an  atmosphere  have 
also  been  observed.     The  mass  of  the  planet,  determined  by 
the  motions  of  its  satellites,  is  about  -g-gVffth  of  that  of  the 
sun,  or  about  93  times  that  of  the  earth. 

217.  Rings  of  Saturn. — When  observed  through  a  tele- 
scope, Saturn  is  seen  to  be  surrounded  by  a  marvelous  system 
of  luminous  rings,  lying  one  within  another  in  the  plane  of  the 
planet's  equator  and  very  nearly  concentric  with  the  planet. 
Although  the  planet  itself  was  known  to  the  ancients,  the 
existence  of  these  rings  was  not  suspected  until  the  seven- 
teenth century.     They  were  then  supposed  to  be  two  rings, 
one  within  the  other;  but  later  observations  show  that  there 


SATURN  197 

are  three  rings,  the  outer  two  are  brightly  luminous  while 
the  inner  one  is  only  feebly  luminous  and  is  semi-transparent. 

The  distance  between  the  outer  ring,  A,  and  the  inner 
luminous  ring,  B,  is  about  1600  miles  while  there  is  no  sharp 
line  of  demarcation  between  ring  B  and  the  dusky  innermost 
ring,  C. 

The  dimensions  of  the  rings  are  approximately  as  fol- 
lows: 

Outer  diameter  of  exterior  ring  A  ....  168,000  miles 

Breadth  of  ring  A 10,000  miles 

Distance  between  ring  A  and  ring  B. .  .  1,600  miles 

Breadth  of  ring  B 16,500  miles 

Breadth  of  ring  C 10,000  miles 

There  is  thus  left  a  clear  space  of  about  10,000  miles 
between  the  inner  edge  of  the  dusky  ring,  C,  and  the  planet's 
equator. 

218.  The  thickness  of  the  rings  is  very  small.  Sir  John 
Herschel  estimated  it  at  not  more  than  250  miles,  while 
Professor  Bond  considered  it  to  be  only  about  40  miles.  The 
rings  appear  to  rotate  about  the  planet  from  west  to  east, 
the  period  of  rotation  being  about  10J  hours,  according  to 
Herschel.  Various  theories  have  been  advanced  as  to  their 
composition.  At  first  they  were  supposed  to  be  solid;  but 
Laplace  demonstrated  that,  if  a  ring  were  homogeneous  and 
solid,  the  smallest  disturbing  force  would  throw  it  out  of 
equilibrium  and  cause  it  to  fall  upon  the  planet.  In  1850 
Professor  Bond  advanced  the  theory  that  the  rings  were  in  a 
fluid  state;  and  Professor  Peirce,  having  shown  that  no  solid 
ring,  regular  or  irregular  in  shape  and  constitution,  could 
remain  in  equilibrium  about  Saturn,  adopted  Professor 
Bond's  theory,  and  also  concluded  that  the  equilibrium  of 
the  rings  was  maintained  by  the  attraction  exerted  by 
Saturn's  satellites. 

All  these  theories  have  been  abandoned.  The  theory 
now  prevailing  is  that  the  rings  are  a  collection  of  tiny 


198  THE  PLANETS  AND  THE  PLANETOIDS 

satellites,  revolving  about  the  planet  precisely  as  the  asteroids 
or  minor  planets  are  found  to  revolve  about  the  sun.  This 
theory  was  first  advanced  by  Cassini,  in  1815,  and  was 
renewed  by  Professor  Clerk  Maxwell  in  1856. 

219.  The  rings  «iust  present  a  magnificent  spectacle  to 
the  inhabitants  of  the  planet.     To  an  observer  on  Saturn's 
equator  they  will  appear  as  an  arch,  passing  through  the 
zenith,  and  through  the  east  and  the  west  point  of  the  horizon. 
To  such  an  observer  only  the  edge  of  the  rings  is  visible. 
As  he  moves  away  from  the  equator,  the  altitude  of  the 
rings  decreases,  and  the  side  of  the  rings  becomes  visible, 
presenting  an  appearance  not  unlike  the  familiar  one  of  the 
rainbow.     Under  the  most  favorable  circumstances  of  posi- 
tion, the  rings  will  be  projected  against  the  sky  as  an  arch 
with   the   enormous   angular   breadth  of  about  15°,  which 
is  about  30  times  the  diameter  which  the  sun  presents 
to  us. 

220.  Disappearance  of  the  Rings. — As  Saturn  revolves 
about  the  sun,  the  plane  of  its  rings  remains,  like  the  plane  of 
the  earth's  equator,  fixed  in  space,  and  intersects  the  plane 
of  the  ecliptic  in  a  line  which  is  called  the  line  of  nodes  of 
the  rings.     In  Fig.  71,  let  S  be  the  sun,  ABCD  the  orbit  of 
the  earth,  and  EHLN  the  orbit  of  Saturn.     Let  HN  be 
the  line  of  nodes  of  the  rings,  and  draw  the  lines  GO  and  KM 
parallel  to  HN,  and  tangent  to  the  earth's  orbit.     When  the 
planet  is  at  H,  the  plane  of  the  rings  passes  through  the  sun, 
and  only  the  edge  of  the  rings  is  illuminated.     In  such  a  case 
the  rings  cannot  be  seen,  or  at  all  events  can  only  be  seen,  in 
very  powerful  telescopes,   as  an  exceedingly  narrow  line. 
Furthermore,  if,  while  the  planet  is  within  the  lines  GO  and 
KM,  the  earth  encounters  the  plane  of  the  rings,  they  will 
not  be  visible.     And  thirdly,  if,  while  the  planet  is  within  the 
same  limits,  the  plane  of  the  rings  passes  between  the  earth 
and  the  sun,  the  dark  side  of  the  rings  will  be  turned  towards 
the  earth,  and  they  will  not  be  seen.     When  the  planet  is 
beyond  these  limits,  it  is  evident  that  the  rings  will  always  be 


SATURN 


199 


visible,  and  will  present  an  elliptical  appearance,  as  repre- 
sented at  E. 

Now,  we  can  readily  compute  the  length  of  time  which 
Saturn  requires  in  passing  through  the  arc  GK.  For  in  the 
triangle  CSK,  right-angled  at  C,  we  know  the  sides  CS  and 
KS,  or  the  distance  of  the  earth  from  the  sun  and  that  of  the 
planet,  and  can  therefore  obtain  the  angle  CKS.  It  will  be 
found  to  be  about  6°  1'.  This  angle  is  equal  to  the  angle 


KSH,  and  therefore  double  this  angle,  or  12°  2',  is  the  angle 
through  which  Saturn  moves  about  the  sun  in  passing  through 
the  arc  GK.  Now  we  know  that  Saturn  makes  a  complete 
revolution  about  the  sun  in  10,759  days;  and  therefore  we 
may  find  by  a  simple  proportion  the  time  which  it  requires  to 
pass  through  12°  2'.  This  time  is  found  to  be  359.6  days,  or 
very  nearly  a  sidereal  year;  so  that  the  earth  makes  very 
nearly  one  complete  revolution  about  the  sun  while  Saturn 
is  passing  through  the  arc  GK. 

221.  Number  of  Disappearances. — Since  Saturn's  period 
of  revolution  is  29.45  years,  these  disappearances  will  occur 


200  THE  PLANETS  AND  THE  PLANETOIDS 

at  intervals  of  a  little  less  than  15  years.  Since  the  time 
during  which  the  planet  remains  within  the  limits  GO  and 
KM  is  only  six  days  less  than  a  year,  and  since  the  earth  may 
encounter  the  plane  of  the  ring  at  any  point  in  its  orbit,  it  is 
quite  certain  that  one  such  meeting  will  occur,  and  under 
certain  circumstances  there  may  be  three.  Suppose,  for 
instance,  that  the  earth  is  at  a  when  Saturn  is  at  G,  and  that 
both  bodies  move  about  the  sun  in  the  direction  EHLN. 
The  earth  will  meet  the  plane  of  the  rings  somewhere  in  the 
arc  a  A,  and  the  rings  will  disappear.  The  rings  will  con- 
tinue to  be  invisible  for  some  time,  since  their  plane  will  lie 
between  the  earth  and  the  sun.  The  earth  will  overtake  the 
plane  before  Saturn  reaches  the  point  H,  and  after  that  time 
the  rings  will  be  visible  until  the  planet  is  at  H,  when  the 
plane  passes  through  the  sun,  and  the  rings  again  disappear. 
The  earth  will  now  be  near  the  point  b  and  the  rings  will 
continue  to  be  invisible,  since  their  dark  side  is  turned  to  the 
earth.  The  earth,  passing  through  C,  will  again  meet  the 
plane  somewhere  in  the  arc  CD,  and  after  that  time  the  rings 
will  be  visible.  No  more  disappearances  will  occur  for 
about  fifteen  years,  at  the  end  of  which  interval  the  planet 
will  pass  through  the  arc  MO. 

The  last  disappearance  took  place  in  1907;  the  next  will 
take  place  in  1922. 

URANUS 

222.  All  the  planets  which  have  thus  far  been  described, 
except  of  course  the  minor  planets,  were  known  to  the 
ancients;  but  the  last  two  are  among  the  comparatively 
recent  discoveries  of  astronomers.  Uranus  was  discovered 
by  Sir  William  Herschel,  in  1781,  by  pure  accident.  Herschel 
as  well  as  other  astronomers  whose  attention  was  directed 
to  it,  at  first  supposed  it  to  be  a  comet;  and  it  was  only  after 
several  months  of  observation  that  it  was  found  to  be  a 
planet.  Several  names  were  suggested  for  it,  but  the  name 


URANUS  201 

of  Uranus  was  finally  adopted.  The  astronomical  symbol 
for  it  which  the  English  have  adopted  is  formed  from  the 
initial  letter  of  Herschel's  name. 

Upon  searching  the  star  catalogues  and  other  astronom- 
ical records,  it  was  found  that  the  planet  had  been  observed 
no  less  than  twenty  times  in  the  preceding  90  years,  and  had 
been  considered  to  be  a  fixed  star,  its  daily  motion  being  so 
slight  as  to  have  escaped  notice.  Indeed  its  period  is  so 
great  that  even  its  annual  change  of  position  is  only  a  few 
degrees. 

223.  These  previous  records,  however,  were  of  great 
assistance  to  astronomers  in  the  determination  of  the  elements 
of  the  planet's  orbit.  The  synodical  period  of  the  planet  is 
370  days,  and  the  sidereal  period  30,687  days,  or  about  84 
years.  Its  distance  from  the  sun  is  1,784,732,000  miles,  and 
its  diameter  30,193  miles.  It  is  barely  visible  to  the  naked 
eye  at  opposition. 

It  is  attended  by  four  satellites,  which  are  only  visible  in 
the  most  powerful  telescopes;  and  by  means  of  their  move- 
ments the  mass  of  the  planet  is  found  to  be  about  15  times 
that  of  the  earth.  One  very  remarkable  point  about  these 
satellites  is  that  their  motion  about  their  primary  is  retro- 
grade, or  from  east  to  west,  in  planes  inclined  about  82° 
to  the  plane  of  the  ecliptic,  while  the  motions  of  most  of  the 
other  satellites  which  have  hitherto  been  described  are  from 
west  to  east,  and  in  planes  making  very  small  angles  with  the 
plane  ot  the  ecliptic.  Sir  Wm.  Herschel  believed  that  he 
had  discovered  two  other  satellites;  but  recent  observations 
with  powerful  telescopes  do  not  confirm  his  belief. 

Scarcely  more  can  be  said  of  the  physical  appearance  of 
Uranus  than  that  it  is  uniformly  bright. 

There  are  sometimes  visible  upon  the  disc  of  Uranus  faint 
bands  or  belts  somewhat  similar  to  those  of  Jupiter,  and 
observations  made  at  Nice  in  1883  under  unusually  favorable 
circumstances  seemed  to  show  that  the  surface  of  Uranus 
is  similar  in  appearance  to  the  surface  of  Mars;  that  is  to 


202  THE  PLANETS  AND  THE  PLANETOIDS 

say,  dark  spots  were  seen  near  the  center  of  the  disc  and 
white  spots  were  seen  near  the  circumference.  The  motion 
of  these  spots  seemed  to  give  the  planet  a  rotation  period 
of  about  ten  hours;  but  it  cannot  be  said  that  any  determina- 
tion of  this  period  of  rotation  is  to  be  trusted. 

NEPTUNE 

224.  Early  in  the  last  century  the  conviction  forced  itself 
upon'  the  minds  of  many  astronomers  that  there  must  exist 
still  another  planet,  exterior  to  Uranus.  The  circumstance 
which  led  to  this  conclusion  was  the  existence  of  irregularities 
in  the  orbit  of  Uranus,  over  and  above  the  irregularities  which 
were  due  to  the  attractions  exerted  by  the  planets  then 
known.  The  first  systematic  attempt  to  deduce  the  elements 
of  the  orbit  of  this  unknown  planet  from  these  irregularities 
seems  to  have  been  made  by  Mr.  Adams,  of  England,  in 
1843-5.  The  position  which  he  assigned  to  the  planet  was  in 
heliocentric  longitude  329°  19',  but  this  determination  was 
not  then  made  public.  The  same  intricate  problem  was 
also  solved  by  M-  Leverrier,  of  Paris,  in  1845-6,  and  the 
longitude  which  he  obtained  was  326°.  During  the  summer 
of  1846  search  was  made  for  the  planet  in  England,  but 
without  success,  owing  to  the  want  of  a  proper  star-map. 
The  observatory  at  Berlin,  however,  was  better  supplied; 
and  on  the  night  of  September  23d,  in  compliance  with  a 
request  made  in  a  letter  received  that  day  from  Leverrier, 
Dr.  Galle  at  once  detected,  in  longitude  326°  52',  what  was 
apparently  a  star  of  the  eighth  magnitude,  though  it  was  not 
laid  down  on  the  map.  Subsequent  observations  showed 
that  this  body  was  really  a  planet,  and  it  was  agreed  to  give 
it  the  name  of  Neptune. 

"Such,"  in  the  words  of  Hind,  "  is  a  brief  history  of  this 
most  brilliant  discovery,  the  grandest  of  which  astronomy 
can  boast,  and  an  astonishing  proof  of  the  power  of  the 
human  intellect." 


NEPTUNE  203 

225.  The  synodical  period  of  Neptune  is  367  days,  and  its 
sidereal  period  60,187  days,  or  about  164  years.     Its  mean 
distance  from  the  sun  is  2,800,000,000  miles,  and  its  diameter 
34,800  miles.     It  is  attended  by  one  satellite,  and  some 
astronomers  suspect  the  existence  of  a  second.     The  mass  of 
Neptune  is  about  seventeen  times  that  of  the  earth.     The 
planet  is  not  visible  to  the  naked  eye. 

Nothing  is  yet  determined  as  to  the  physical  appearance 
or  the  axial  rotation  of  the  planet.  A  remarkable  circum- 
stance in  connection  with  the  satellite  is  that,  like  the  satel- 
lites of  Uranus,  it  moves  about  its  primary  from  east  to  west. 

226.  It  may  help  us  in  our  conception  of  the  immense 
distance  of  Neptune  from  us,  even  when  it  is  in  opposition, 
to  consider  that  light,  with  its  velocity  of  186,000  miles  a 
second,  requires  four  hours  to  come  from  the  planet  to  the 
earth.     If  there  are  any  inhabitants  of  Neptune,  the  sun  will 
to  them  have  an  apparent  diameter  of  only  wth  of  what  it 
has  to  us,  since  the  distance  of  Neptune  from  the  sun  is 
about  thirty  times  that  of  the  earth.     It  will  therefore  appear 
to  them  only  about  as  large  as  Venus  appears  to  us,  under 
the   most   favorable   circumstances.     Saturn,    Jupiter,  and 
Uranus  may  possibly  be  visible  to  them  as  extremely  small 
bodies,  but  it  is  very  doubtful  if  any  of  the  other  planets  of 
our  system  are  visible,  even  with  the  strongest  telescopes. 

227.  Relative  Sizes  and  Distances  of  the  Planets. — The 
relative  distances  of  the  planets  from  the  sun,  their  relative 
magnitudes,  as  well  as  other  numerical  data  concerning  them, 
will  be  found  in  tables  in  the  Appendix.     In  Plate  I.  will  be 
seen  a  representation  of  their  relative  magnitudes,  as  they 
would  appear  to  an  observer  stationed  at  the  same  distance 
from  all  of  them. 

THE  NEBULAR  HYPOTHESIS 

228.  Points  of  Resemblance  in  the  Planetary  Phenomena. 

— The  light  of  the  planets  and  the  satellites,  when  examined 


204  THE  PLANETS  AND  THE  PLANETOIDS 

in  the  spectroscope,  produces  only  the  ordinary  spectrum  of 
reflected  solar  light.  While,  therefore,  the  spectral  analysis 
of  the  light  of  the  sun,  the  stars,  and  other  heavenly  bodies 
which  shine  by  their  own  light,  enables  us  to  determine  to 
some  extent  the  elements  of  which  they  are  composed,  a 
similar  experiment  tells  us  nothing  of  the  constitution  of  the 
planets  or  the  satellites.  What  we  do  know,  however,  of 
their  form,  their  appearance,  their  mass,  and  their  density, 
leads  us  to  conclude  that  they  are  bodies  not  dissimilar  to 
the  earth  in  general  constitution.  There  are,  besides,  certain 
remarkable  coincidences  in  the  various  phenomena  exhibited 
by  the  sun,  the  planets,  and  the  satellites,  which  seem  to 
point  to  a  common  origin  of  the  whole  solar  system.  The 
principal  of  these  coincidences  are  the  following: 

(1)  All  the  planets  revolve  about  the  sun  in  the  same 
direction  in  which  the  sun  rotates  upon  its  axis:  that  is  to 
say,  from  west  to  east. 

(2)  The  planes -of  the  planetary  orbits  nearly  coincide 
with  the  plane  of  the  sun's  equator. 

(3)  The  satellites  of  each  planet,  as  far  as  known,  revolve 
about  their  primary  in  the  same  direction  in  which  the 
primary  rotates  upon  its  axis.     The  satellites  of  Uranus  and 
Neptune  may  or  may  not  form  an  exception  to  this  rule,  for 
these  planets  are  so  distant  that  observation  fails  to  make 
certain  the  axial  rotation  of  them. 

(4)  The  planes  of  the  orbits  of  the  satellites  of  each 
planet  approximately  coincide  with  the  plane  of  that  planet's 
equator. 

(5)  Both  planets  and  satellites  revolve  in   ellipses  of 
small  eccentricity. 

229.  The  Nebular  Hypothesis. — The  idea  of  the  nebular 
hypothesis  seems  to  have  presented  itself  at  about  the  same 
time  to  both  Sir  William  Herschel  and  Laplace.  The 
principal  points  in  it  are  the  following.  All  the  matter  which 
now  composes  the  sun,  the  planets,  and  the  satellites  once 
existed  as  a  single  nebulous  mass,  extending  beyond  the 


THE  NEBULAR  HYPOTHESIS  205 

present  orbit  of  Neptune,  and  rotating  on  an  axis  from  west 
to  east.  In  the  progress  of  ages  this  nebulous  mass  slowly 
contracted  and  condensed,  from  the  loss  of  the  heat  which  it 
radiated  into  space,  and  from  the  gravitation  of  its  particles 
towards  the  center.  As  its  dimensions  became  less,  its 
velocity  of  rotation  became  greater,  according  to  the  laws  of 
Mechanics :  since  any  particle  moving  in  a  circle  of  any  radius 
with  a  certain  linear  velocity  would,  as  it  approached  the 
center,  move  in  a  smaller  circle  with  nearly  the  same  linear 
velocity,  and  would  therefore  have  a  greater  angular  velocity. 
Finally,  the  centrifugal  force  generated  by  this  increased 
velocity  at  the  surface  of  the  equator  of  the  mass  exceeded  the 
attraction  towards  the  center,  and  a  nebulous  zone  was 
detached,  which  revolved  independently  of  the  interior  mass, 
just  as  the  rings  of  Saturn  have  been  seen  to  revolve  about  that 
planet.  This  zone,  by  concentration  at  certain  points  within 
itself,  broke  up  into  separate  masses;  and  these  masses, 
either  from  slight  differences  of  velocity  or  from  the  pre- 
ponderating attraction  of  some  faction  larger  than  the 
others,  eventually  formed  one  body,  revolving  about  the 
central  mass.  And,  furthermore,  as  these  separate  masses 
came  together,  a  motion  of  rotation  was  communicated  to 
the  combined  mass,  just  as  a  whirlpool  or  an  eddy  is  formed 
when  two  streams  of  water  meet;  and  this  rotating  mass, 
condensing  and  contracting  in  its  turn,  threw  off  from  itself  a 
second  zone,  which  underwent  all  the  changes  above  de- 
scribed. Thus  were  formed  a  planet  and  its  satellite,  each 
revolving  about  its  primary  in  the  direction  of  that  primary's 
axial  rotation:  and  by  a  continuation  of  the  process  the 
whole  system  of  planets  and  satellites  was  evolved. 

230.  Necessary  Conditions. — It  is  a  necessary  condition 
of  the  truth  of  this  hypothesis,  that  the  planets  shall  revolve 
(as  they  do  revolve)  about  the  sun  in  the  same  direction  in 
which  it  rotates.  It  is  also  necessary  that  each  satellite  or 
system,  of  satellites  shall  revolve  about  its  primary  in  the 
same  direction  in  which  that  primary  rotates.  It  is  not, 


206  THE  PLANETS  AND  THE  PLANETOIDS 

however,  absolutely  necessary  that  the  outer  planets  shall 
rotate  in  the  same  direction  in  which  they  revolve;  although 
such  a  coincidence  might  be  expected,  since  the  revolution  of 
the  outer  particles  from  which  a  planet  was  formed  would  be 
more  rapid  than  that  of  those  which  were  nearer  to  the  sun. 
If  we  assume  this  hypothesis  to  be  true,  the  rings  of 
Saturn  are  to  be  considered  as  rings  which  did  not  form 
satellites  after  they  were  thrown  off  from  the  planet;  while 
in  the  case  of  the  minor  planets  the  ring  broke  up  into^sepa- 
rate  masses,  which  have  continued  to  revolve  in  independent 
orbits  about  the  sun. 

231.  Experiment  in  Support  of  the  Hypothesis. — The 
possible  truth  of  the  nebular  hypothesis  is  supported  by  an 
ingenious  experiment  devised  by  M.  Plateau.*     A  mass  of 
olive-oil  was  immersed  in  a  mixture  of  alcohol  and  water, 
the  density  of  the  mixture  being  made  exactly  equal  to  that 
of  the  oil.     In  this  way  the  mass  of  oil  was  practically  with- 
drawn from  the  influence  of  gravitation.     When  made  to 
rotate,  the  mass  assumed  a  spheroidal  form,  and  finally, 
when  the  velocity  of  rotation  was  sufficiently  great,  a  ring  of 
matter  was  thrown  off  in  the  equatorial  region.     This  ring 
subsequently  broke  up  into  independent  masses,  each  of 
which  assumed  a  globular  form,  rotated  on  an  axis  of  its  own, 
and  continued  to  revolve  about  the  central  mass:  thus  pre- 
senting   precisely    the    successive    phenomena    which    are 
assumed  in  the  nebular  hypothesis  to  have  occurred  in  the 
formation  of  the  solar  system. 

232.  The  truth  of  the  nebular  hypothesis  is  by  no  means 
universally  admitted  by  astronomers  and  other  scientific 
men;    and  it  is  difficult  to  say  what  is  the  predominant 
belief  about  it  at  the  present  time.     The  high  scientific 
reputation  of  those  who  originated  it,  and  of  those  who  have 
since  supported  it,  is  sufficient  justification  for  giving  it  a 

*  See  Annales  de  Chimie,  vol.  xxx.  (1850).  The  experiment  is  also 
described  in  Carpenter's  Mechanical  Philosophy,  etc.,  one  of  the  volumes 
of  Bohn's  Scientific  Library  (London). 


THE  NEBULAR  HYPOTHESIS  207 

place  in  this  treatise;  but  it  must  not  be  forgotten  that  its 
truth  is  still  very  emphatically  an  open  question,  and  that 
many  great  minds  are  numbered  with  its  opponents. 

Sir  William  Herschel  was  led  to  the  adoption  of  the 
nebular  theory  by  his  examination  of  that  class  of  celestial 
bodies  called  nebulae,  some  of  which  presented  in  his  day, 
and  present  now,  the  appearance  of  masses  of  nebulous 
matter.  Recent  spectroscopic  examinations  of  some  of  these 
nebulae  (Art.  286)  go  to  show  that  they  are  really  what  they 
seem  to  be,  masses  of  incandescent  vapor;  and  this  discovery 
gives  a  new  interest  to  the  nebular  hypothesis.  Mr.  Lock- 
yer,  in  his  Elementary  Lessons  in  Astronomy,  says  that  "it 
may  take  long  years  to  prove  or  disprove  this  hypothesis; 
but  it  is  certain  that  the  tendency  of  recent  observations  is 
to  show  its  correctness." 

Fresh  doubts  are  thrown  upon  the  truth  of  the  nebular 
hypothesis  by  the  discovery  of  the  satellites  of  Mars  (§  203) ; 
since  the  angular  velocity  of  the  inner  satellite  appears  to  be 
three  times  as  great  as  the  rotation  of  the  planet. 

A  theory  of  the  origin  of  the  solar  system  which  is  gain- 
ing ground  among  astronomers  is  known  as  the  "Planetesimal 
Hypothesis."  According  to  this  theory  a  star  of  mass  several 
times  greater  than  the  sun  approached  it  so  closely  that  its 
tidal  action  distorted  the  sun  more  than  was  consistent  with 
stability,  and  either  one  or  two  streams  of  matter  were  shot 
out  from  the  sun  with  considerable  velocity.  These  streams 
were  longitudinally  unstable  and  broke  up  almost  at  once 
into  a  series  of  fluid  masses.  These  masses  are  called  plane- 
tesimals  and  the  larger  ones,  increased  gradually  by  the  accre- 
tion of  other  smaller  ones,  formed  the  planets.  The  satellites 
are  smaller  plane tesimals  shot  out  near  the  larger  ones  that 
became  planets. 


CHAPTER  XIII 
COMETS  AND  METEORIC  BODIES 

COMETS 

233.  General  Description  of  Comets. — A  comet  is  a  body 
of  nebulous  appearance  and  irregular  shape,  revolving  in  an 
orbit  about  the  sun.  Comets  have  usually  been  considered 
to  consist  for  the  most  part  of  nebulous  matter;  but  the  theory 
has  lately  been  advanced  that  they  are  collections  of  minute 
meteoric  bodies  surrounded  by  atmospheres  of  incandescent 
gas. 

Comets  differ  widely  from  each  other  in  appearance,  and 
no  description  of  them  can  be  given  to  which  there  will  not  be 
many  exceptions.  Generally  speaking,  a  comet  consists  of 
three  parts:  the  nucleus,  the  coma,  and  the  tail.  The  nucleus 
and  the  coma  together  form  the  head.  The  nucleus  is  a 
bright  point,  like  a  star  or  a  planet,  which  may  be  either  a 
solid  mass,  or  a  mass  of  nebulous  matter  of  a  density  greater 
than  that  of  the  rest  of  the  comet.  The  diameter  of  the 
nucleus  varies  considerably  in  different  comets:  that  of  the 
comet  of  1845  (iii)  *  was  about  8000  miles,  while  that  of  the 
comet  of  1806  was  only  30  miles.  The  average  value  is  not 
over  500  miles;  and  in  many  comets  no  nucleus  whatever  is 
perceptible. 

The  coma  is  a  mass  of  cloud-like  matter,  more  or  less 
nearly  globular  in  form,  which  surrounds  the  nucleus.  The 
nucleus,  however,  as  a  general  thing,  is  not  situated  at  the 
center  of  the  coma,  but  lies  towards  that  margin  which  is  the 

*  The  number  (iii)  means  that  this  was  the  third  comet  which 
appeared  in  the  course  of  the  year. 

208 


COMETS  209 

nearer  to  the  sun.  The  diameter  of  the  coma  is  different  in 
different  comets:  that  of  the  comet  of  1847  (v)  was  only 
18,000  miles,  while  that  of  the  comet  of  1811  (i)  was  over 
1,000,000  miles.  Usually,  however,  it  is  less  than  100,000 
miles.  It  is  frequently  noticed  that  the  coma  decreases  in 
apparent  diameter  as  the  comet  approaches  the  sun,  and  in- 
creases as  the  comet  recedes  from  it.  On  the  supposition 
that  the  coma  consists  of  vaporous  matter,  this  phenomenon 
is  explained  by  the  assumption  that  the  intense  heat  to 
which  the  comet  is  subjected  as  it  approaches  the  sun  is 
sufficient  to  rarefy  this  vaporous  matter  to  such  a  extent 
that  some  of  it  becomes  invisible. 

The  tail  is  a  train  of  cloud-like  matter  attached'  to  the 
head,  which  usually  lies  in  a  direction  nearly  opposite  to  that 
in  which  the  sun  lies  from  the  head.  The  tail  is  usually  very 
small  when  the  comet  first  appears,  and  sometimes  is  not 
even  perceptible.  As  the  comet  approaches  the  sun,  the 
length  of  the  tail  increases,  and  sometimes  becomes  enormous. 
In  the  comet  of  1811  (i),  for  instance,  the  length  of  the  tail 
was  100,000,000  miles;  and  in  that  of  1843  (i)  it  was  200,000- 
000  miles. 

The  angular  length  of  the  tail  depends  not  only  on  its 
absolute  length,  but  also  on  its  distance  from  the  earth,  and 
on  the  direction  in  which  the  axis  of  the  tail  lies.  There  are 
six  comets  on  record  of  which  the  tails  subtended  angles  of 
over  90°;  and  one  of  these,  that  of  1861  (ii),  had  a  tail  of 
104°  in  length,  as  observed  at  some  places. 

234.  Diversity  of  Appearance. — The  description  above 
given  may  be  considered  to  apply  to  comets  taken  as  a 
class;  but,  as  already  remarked,  important  exceptions  are 
often  noticed  in  individual  comets.  Indeed,  it  is  hardly 
possible  to  compare  any  two  comets  without  finding  marked 
points  of  difference  in  them.  Some  comets  are  not  visible 
at  all,  except  by  the  aid  of  powerful  telescopes,  and  are  hence 
called  telescopic  comets;  while  others,  again,  are  so  conspicuous 
as  to  be  visible  to  the  naked  eye  in  full  daylight.  Some 


210  COMETS  AND  METEORIC  BODIES 

comets  have  more  than  one  tail;  the  comet  of  1823,  for  in- 
stance, had  a  tail  turned  towards  the  sun,  in  addition  to  the 
usual  one  turned  from  it.  The  comet  of  1744  is  reported 
to  have  had  six  tails,  spread  out  like  an  immense  fan,  through 
an  angle  of  117°;  but  the  truth  of  the  record  is  not  above 
suspicion. 

Not  only  do  comets  differ  thus  widely  from  each  other  in 
appearance,  but  even  the  same  comet  changes  its  appearance 
from  day  to  day.  Sometimes  the  nucleus  decreases  in 
diameter  as  it  approaches  the  sun:  sometimes  its  brightness 
increases,  and  jets  of  luminous  matter  are  thrown  off  from  it 
in  the  direction  of  the  sun.  The  length  of  the  tail  often 
increases  with  marvelous  rapidity;  in  the  case  of  the  Great 
Comet  of  1843  (i),  the  increase  was  estimated  to  be  about 
35,000,000  miles  a  day,  after  the  comet  had  passed  its 
perihelion.  There  are  some  instances  on  record  of  a  comet's 
having  separated  into  two  distinct  comets.  This  is  asserted 
in  the  Greek  records  of  a  comet  which  appeared  in  370  B.C., 
and  Biela's  comet  presents  an  indubitable  instance  of  this 
kind.  This  comet  was  observed  in  1826  and  1832,  and  was 
determined  to  be  a  comet  with  a  period  of  nearly  seven 
years.  Its  return  in  1839  was  not  observed.  It  again 
appeared  in  1846,  and  then  presented  the  extraordinary 
appearance  of  two  comets,  moving  side  by  side,  at  a  distance 
apart  of  over  150,000  miles. 

At  the  next  reappearance  in  1852  the  two  parts  were 
much  more  widely  separated  and  since  that  time  the  comet 
has  not  reappeared. 

235.  The  Tail.— The  general  form  of  the  tail  is  that  of  a 
truncated  cone,  the  larger  base  being  at  the  extremity  of  the 
tail.  It  is  noticed  that  the  tail  is  always  brighter  near  the 
borders  than  along  the  middle,  from  which  it  is  inferred  that 
it  is  hollow:  since  only  on  such  a  supposition  would  the  line 
of  sight  pass  through  more  luminous  matter  when  directed 
to  the  edges  than  when  directed  to  the  middle.  With  regard 
to  the  formation  of  the  tail,  the  most  generally  accepted 


COMETS 


211 


theory  seems  to  be  that  the  matter  of  which  the  nucleus  is 
composed  is  excited  and  dilated  by  the  action  of  the  sun's 
rays,  as  the  comet  approaches  the  sun,  and  that  particles  of 
vaporous  matter  are  thrown  off  from  it;  and  that  these 
particles  are  driven  to  the  rear  by  some  repulsive  force  exerted 
by  the  sun,  and  thus  form  the  tail.  What  this  repulsive 
force  exerted  by  the  sun  is,  has  not  yet  been  determined;  but 
the  general  situation  which  the  tail  of  a  comet  has  with  refer- 
ence to  the  sun  seems  to  show  that  some  such  force  (perhaps 
radio-active,  perhaps  the  pressure  of  light)  exists.  Nor  has 
it  yet  been  determined  what  is  the  force  which  originally 
detaches  these  vaporous  particles  from  the  nucleus:  it  may 
be  the  same  repelling  force  which  drives  them  to  the  rear, 
it  may  be  a  force  generated  in  the  nucleus  itself,  or  it  may 
be  a  combination  of  both 
these  forces.  If  we  adopt 
the  theory  of  the  meteoric 
structure  of  these  bodies, 
the  tail  is  to  be  considered 
as  a  cloud  of  minute  par- 
ticles of  matter,  held  to- 
gether by  their  mutual 
attraction,  or  by  the  at- 
traction exerted  upon 
them  by  the  denser  mass 
which  constitutes  the 
head. 

236.  Curvature  of  the  Tail. — The  tail  of  a  comet  is 
usually  not  straight,  but  is  concave  towards  that  part  of 
space  which  the  comet  is  leaving.  If  we  assume  the  existence 
of  a  solar  repulsive  force,  similar  to  that  mentioned  in  the  pre- 
ceding article,  this  peculiarity  of  shape  may  be  thus  explained. 
In  Fig.  72,  let  S  be  the  sun,  and  GCD  a  portion  of  the  orbit  of 
a  comet.  When  the  nucleus  is  at  A,  let  a  particle  be  driven 
from  it  in  the  direction  S  A ,  with  a  force  sufficient  to  carry  it 
to  L  in  the  time  in  which  the  nucleus  moves  from  A  to  C. 


FIG.  72. 


212  COMETS  AND  METEORIC  BODIES 

When  the  nucleus  reaches  C,  this  particle,  still  retaining  the 
motion  which  it  had  in  common  with  the  nucleus,  will  be 
found  at  some  point  M .  In  the  same  way  a  particle  driven 
from  the  nucleus  when  it  is  at  B  will  be  found  at  some  point 
K,  when  the  nucleus  reaches  C:  and,  in  general,  when  the 
nucleus  is  at  C  the  tail  will  not  lie  in  the  direction  SN,  but 
in  the  direction  of  the  curve  CKM,  as  shown  in  the  figure. 

237.  Elements  of  a  Comet's  Orbit. — A  comet  is  identified 
at  its  successive  returns,  not  by  its  appearance,  which  is 
liable,  as  we  have  already  seen,  to  serious  changes,  but  by  the 
elements  of  its  orbit.     In  consequence  of  the  comparative 
ease  with  which  the  elements  of  a  parabola  can  be  calculated, 
astronomers  are  in  the  habit  of  using  that  curve  to  represent 
at  first  the  approximate  form  of  a   comet's  orbit.     The 
elements  of  a  parabolic  orbit  are  five  in  number,  and  are  as 
follows : 

(1)  The  inclination  of  the  orbit  to  the  plane  of  the 
ecliptic. 

(2)  The  longitude  of  the  ascending  node. 

(3)  The  longitude  of  the  perihelion. 

(4)  The  time  at  which  the  comet  passes  its  perihelion. 

(5)  The  distance  of  the  comet  from  the  sun  at  perihelion. 
Tables  and  formulae  have  been  constructed  by  which  these 
elements  can  be  computed  from  the  results  of  three  distinct 
observations  of  the  position  of  the  comet:  and  these  three 
observations  may  all  be  made,  if  necessary,  within  the  space 
of  48  hours.     The  parabolic  elements  having  thus  been  ob- 
tained, the  catalogues  of  comets  are  searched  to  see  if  these 
elements  are  similar  to  those  recorded  of  any  previous  comet. 
As  it  is  highly  improbable  that  the  elements  of  any  two 
comets  will  coincide  throughout,  the  presumption  is  a  strong 
one,  if  two  comets,  visible  at  different  times,  move  in  the  same 
orbit,  that  they  are  one  and  the  same  comet:  and  the  more 
often  the  coincidence  is  repeated,  the  more  nearly  does  the 
presumption  approach  to  a  demonstration. 

238.  Number  of  Comets,  and  their  Orbits.— The  number 


COMETS  213 

of  comets  which  have  been  recorded  since  the  Christian  era 
is  about  850:  and  there  are  about  80  recorded  as  observed 
before  that  date.  Of  these  930  appearances  of  comets,  some 
may  undoubtedly  have  been  only  reappearances  of  the  same 
comet:  and,  indeed,  in  some  cases  comets  have  been  identified 
with  other  comets  previously  observed;  but  this  can  hardly 
be  the  case  with  the  majority  of  these  bodies.  Besides  these 
comets  thus  recorded,  there  must  have  been  many  others  so 
situated  as  to  be  above  the  horizon  only  in  the  day-time :  and 
such  comets  would  become  visible  only  in  case  of  the  occur- 
rence ©f  a  total  solar  eclipse.  A  coincidence  of  this  kind  is 
recorded  by  Seneca  as  having  occurred  62  B.C.,  when  a 
large  comet  was  seen  in  close  proximity  to  the  sun  during  a 
solar  eclipse.  The  improvement  of  telescopes  in  recent 
years  has  greatly  increased  the  number  of  comets  which  be- 
come visible,  and  204  were  observed  between  the  years  1800 
and  1876.  We  are  justified,  therefore,  in  concluding  that 
the  comets  which  have  really  come  within  our  system  since 
the  Christian  era  are  to  be  reckoned  by  thousands.  Two 
centuries  and  more  ago,  Kepler  made  the  remarkable  state- 
ment that  "there  are  more  comets  in  the  heavens  than  fishes 
in  the  ocean." 

The  orbit  in  which  a  comet  moves  may  be  either  an  ellipse, 
a  parabola,  or  an  hyperbola.  From  computations  of  the  orbits 
of  398  comets  subjected  to  mathematical  investigations  the 
following  results  may  be  tabulated: 

Comets  with  elliptical  orbits 27 

Subsequent  returns  of  these  comets 81 

Comets  with  elliptical  orbits,  which  have  not 

returned 67 

Comets  with  parabolic  orbits 217 

Comets  with  hyperbolic  orbits 6 

A  comet  whose  orbit  is  either  a  parabola  or  an  hyperbola  will 
not  return  to  our  system;  provided,  at  least,  that  the  attrac- 
tion of  other  bodies  does  not  alter  the  character  of  the  orbit. 


214  COMETS  AND  METEORIC  BODIES 

It  must  be  noticed,  however,  that  some  of  the  orbits  which 
are  called  parabolic,  may  really  be  ellipses  of  an  eccentricity 
so  great  as  to  render  their  elements  unclistinguishable  from 
those  of  parabolas.  In  whatever  conic  section  a  comet  may 
move,  the  sun  is  always  at  the  focus. 

239.  Periodic  Times. — The  90  comets  which  have  been 
found  to  move  in  elliptical  orbits  differ  widely  from  each  other 
in  the  length  of  their  periods.     Among  the  27  comets  whose 
returns  have  been  observed,  there  are  18  with  short  periods, 
lying  between  three  and  fourteen  years  and  4  having  periods 
between    sixty    and   eighty    years.      There    is    no    doubt 
that  these  22  comets  are  periodic;  but  there  is  some  uncer- 
tainty with  regard  to  some  others  of  the  remaining  5.     Two 
elements  of  such  uncertainty  are  the  unsatisfactory  character 
of  the  records  of  the  observations  made  in  the  earlier  ages, 
and  the  length  of  time  which  the  periods  embrace,  being 
often    several   hundred   years.     Halley's    comet,    however, 
Olber's  comet  and  the  comet  of  1812,  with  periods  of  seventy- 
six,  seventy-three  and  seventy-two  years  respectively  are 
included  in  the  list  of  periodic  comets.     There  are  3  other 
comets,  with  periods  not  very  different  from  that  of  Halley's, 
which  were  discovered  within  the  last  century,  and  which 
have  as  yet  made  no  return.     With  regard  to  the  remaining 
comets  to  which  elliptical  orbits  have  been  assigned,  little 
more  can  be  said  than  that  their  periods  embrace  hundreds 
and  even  thousands  of  years. 

240.  Motion  of  Comets  in  their  Orbits. — The  motions  of 
comets  in  their  orbits  about  the  sun  are  not  performed  in  the 
same  direction,  the  number  of  those  whose  motion  is  retro- 
grade being  about  the  same  as  the  number  of  those  whose 
motion  is  direct.     According  to  Chambers,  an  examination  of 
the  motions  of  the  various  comets  shows  "that  with  comets 
revolving  in  elliptic  orbits  there  is  a  strong  and  decided 
tendency  to  direct  motion.     The  same  obtains  with  the 
hyperbolic  orbits:  with  the  parabolic  orbits  there  is  a  rather 
large   preponderance  the   other  way;   and  taking  all  the 


COMETS  215 

calculated  comets  together,  the  numbers  are  too  nearly  equal 
to  afford  any  indication  of  the  existence  of  a  general  law 
governing  the  direction  of  motion." 

The  angles  which  the  planes  of  the  orbits  make  with  the 
plane  of  the  ecliptic  have  values  ranging  from  0°  to  90°;  but 
"there  is  a  decided  tendency  in  the  periodic  comets  to  revolve 
in  orbits  but  little  inclined  to  the  plane  of  the  ecliptic";  and 
if  we  take  all  the  comets  into  consideration,  "we  find  a 
decided  disposition  in  the  orbits  to  congregate  in  and  around 
a  plane  inclined  50°  to  the  ecliptic." 

Owing  to  the  great  eccentricity  of  the  orbits,  some  of  the 
comets  approach  very  near  to  the  sun  at  the  time  of  perihe- 
lion passage:  the  comet  of  1843  (i),  for  instance,  came  within 
100,000  miles  of  the  sun's  surface.  For  the  same  reason  the 
distances  to  which  some  of  the  comets  with  elliptical  orbits 
recede  from  the  sun  are  immense;  thus  the  comet  of  1844 
(ii)  receded  to  a  distance  of  400,000,000,000  miles,  over  130 
timers  the  distance  of  Neptune  from  the  sun.  The  velocity 
of  the  comets  at  perihelion  is  sometimes  enormous;  this  same 
comet  of  1843  swung  about  the  sun  through  an  arc  of  180° 
in  only  two  hours,  and  moved  with  the  velocity  of  350  miles  a 
second. 

241.  Mass  and  Density  of  the  Comets. — The  minuteness 
of  the  mass  of  the  comets  is  proved  by  the  fact  that  they  exert 
no  perceptible  influence  on  the  motions  of  the  planets  or  the 
satellites  although  they  sometimes  pass  very  near  to  them. 
Thus  Lexell's  comet,  1770  (i),  in  its  advance  towards  the  sun, 
became  entangled  with  the  satellites  of  Jupiter,  and  remained 
near  them  for  five  months,  without  sensibly  affecting  their 
motions.  The  effect  of  Jupiter's  attraction  on  the  comet, 
however,  was  very  striking.  The  comet  had  a  period  of 
about  five  years,  and  yet  it  never  appeared  after  1770.  It 
was  found,  by  computation,  that  at  its  first  return  after  that 
date  it  was  so  situated  as  not  to  become  visible;  and  that  in 
1779,  before  its  second  return,  it  came  nearer  to  Jupiter  than 
Jupiter's  fourth  satellite:  and  the  presumption  is  that  its 


216  COMETS  AND  METEORIC  BODIES 

orbit  was  so  changed  and  enlarged  that  the  comet  no  longer 
comes  near  enough  to  the  earth  to  become  visible.  This  same 
comet  came  within  about  1,400,000  miles  of  the  earth  in 
1770:  near  enough,  had  its  mass  been  equal  to  that  of  the 
earth,  to  increase  the 'length  of  the  year  by  nearly  three 
hours;  but  no  sensible  effect  was  produced. 

The  mass,  then,  of  the  comets  being  so  small,  and  their 
volume  so  large,  the  density  of  the  matter  of  which  they  are 
composed  must  be  exceedingly  rare.  Indeed,  it  must  be 
vastly  more  rare  than  that  of  the  lightest  gas  or  vapor  of 
which  we  have  any  knowledge:  for  stars  of  the  smallest 
magnitude  are  distinctly  seen,  and  usually,  too,  with  no 
perceptible  diminution  of  brightness,  through  all  parts  of 
the  comets  excepting  perhaps  the  nucleus;  and  this  too  in 
cases  where  the  volume  of  nebulous  matter  has  a  diameter  of 
50,000  or  100,000  miles. 

242.  Light  of  the  Comets. — The  question  whether  or  not 
comets  shine  by  their  own  light  does  not  seem  to  be  satisfac- 
torily decided.  The  existence  of  phases  would  of  course 
prove  that  they  shine  by  reflected  light;  but  although  in  one 
or  two  cases  the  statement  has  been  made  that  phases  have 
been  detected,  the  truth  of  the  statement  has  in  no  case  been 
universally  accepted.  Undoubtedly  the  distance  of  some 
comets  is  so  great  that  phases  might  exist  and  still  escape 
observation,  as  in  the  case  of  the  superior  planets;  but,  on 
the  other  hand,  some  comets  come  so  near  to  the  earth  that 
there  seems  to  be  no  good  reason  why  phases,  if  any  exist, 
should  not  be  noticed.  Observations  upon  the  light  of  the 
comets  have  been  made,  both  with  the  polariscope  and  with 
the  spectroscope.  The  observations  made  with  the  polar- 
iscope seem  to  establish  the  fact  that  the  comets  shine,  partly 
at  all  events,  by  the  reflected  light  of  the  sun ;  as,  for  instance, 
the  observations  of  Airy  and  others  on  Donati's  comet  in 
1858.  Mr.  Huggins,  of  England,  to  whom  we  owe  so  many 
interesting  discoveries  made  with  the  spectroscope,  has 
recently  examined  the  light  of  several  comets  with  that 


COMETS 


217 


instrument.  In  Brorsen's  comet  he  found  that  the  nucleus 
and  part  of  the  coma  shone  by  their  own  light.  In  Tempers 
comet  the  nucleus  shone  by  its  own  light,  and  the  coma  by 
the  reflected  rays  of  the  sun.  In  some  comets  the  light  from 
the  nucleus  resembled  that  which  comes  from  the  gaseous 
nebulae  In  one  comet  there  was  a  remarkable  resemblance 
in  the  spectrum  produced  by  its  light  to  that  produced  by 
carbon,  not  only  in  the  position  of  the  lines,  but  in  character 
and  relative  brightness. 


LIST  OF  PERIODIC  COMETS  OBSERVED  AT  MORE  THAN 
ONE  RETURN 


Designation. 

1st  Perihelion 
Passage. 

Last  Perihelion  Pass. 
Obs.  up  to  1918. 

Period 
Yrs. 

Least 
Dist. 
Astr.  U. 

Halley  .7  

1456  June    8 

1910  Apr.  19 

75.9 

0.58 

Biela..  

1772  Feb.  16 

1852  Sept.  23 

6.67 

0.98 

Encke  

1786  Jan.   30 

1918  Mar.  24 

3.29 

0.34 

Tuttle  or 

Mechain  .    . 

1790  Jan.   30 

1912  Oct.  28 

13.78 

1.03 

Pons.      

1812  Sept  15 

1884  Jan.    25 

71.36 

0.78 

Olbers  

1815  Apr.  26 

1887  Oct.     8 

73.32 

1.21 

Winnecke   .  . 

1819  July    18 

1915  Sept.    1 

5  67 

0.77 

Faye  

1843  Oct.    17 

1910  Nov.    1 

7.50 

1.69 

DeVico  J 

1844  Sept.    2 

1894  Oct.    12 

5.66 

1.19 

Brorsen    ...    -.'• 

1846  Feb    11 

1890  Feb.  24 

5.52 

0.65 

D  Arrest. 

1851  July     8 

1910  Sept.  16 

6.56 

1.17 

Westphal  

1852  Oct.    12 

1913  Nov.  26 

61.12 

1.26 

Tempel  I  ,  .  .  . 

1867  May  23 

1879  May     7 

5.84 

1.56 

Tempel  Swift 

1869  Nov.  18 

1908  Oct.      1 

5.51 

1.06 

Tempel  11  

1873  June  25 

1915  Apr.   13 

5.28 

1.34 

Wolf           ... 

1884  Nov.    7 

1917  June  16 

6.80 

1.59 

Finlay    

1886  Nov.  22 

1906  Sept.    8 

6.64 

0.99 

Brooks  

1889  Sept.  30 

1911  Jan.     8 

7.10 

1.95 

Holmes  

1892  June  13 

1908  Mar.  14 

6.89 

2.14 

Perrine 

1896  Nov.  24 

1909  Nov.    1 

6.45 

1.17 

Giocobini  

1900  Nov.  28 

1913  Nov.    2 

6.51 

0.97 

Borrelly  

1905  Jan.    16 

1918  Nov.  16 

6.93 

1.40 

218  COMETS  AND  METEORIC  BODIES 

Observations  still  more  recent,  particularly  those  based 
on  photography,  seem  to  show  that  the  light  is  partly  reflected 
sunlight,  and  is  partly  caused  by  the  incandescence  of  the 
comet's  material  and  perhaps  by  radio-activity. 

PERIODIC  COMETS 

243.  It  has  already  been  stated  that  there  are  at  least  22 
comets  which  are  periodic,  18  of  these  being  comets  of  short 
periods,  and  the  other  four  being  Halley's,  Giber's,  West- 
phaPs  and  the  comet  of  1812.     A  list  of  these  comets  will 
be  found  in  the  table  on  page  217,  complete  to  1918. 

ENCKE'S  COMET 

244.  On  November  26,  1818,  a  small  and  ill-defined  tele- 
scopic comet  was  discovered  in  the  constellation  Pegasus,  by 
the  astronomer  Pons,  at  Marseilles.     It  remained  visible 
for  seven  weeks,  and  many  observations  were  made  upon  it. 
Professor  Encke,  of  Berlin,  finding  that  the  elements  of  the 
orb  t  did  not  agree  with  those  of  a  parabola,  determined  to 
subject  them  to  a  rigorous  investigation  according  to  the 
method  proposed  by  Gauss.     This  investigation  showed  that 
the  orbit  was  elliptical,  and  that  the  period  of  the  comet  was 
about  3J  years.     He  further  identified  the  comet  with  the 
comets  of  1786  (i),  1795,  and  1805,  and  predicted  that  it 
would  return  to  perihelion  on  May  24,   1822,  after  being 
retarded  about  nine  days  by  the  influence  of  Jupiter. 

"So  completely  were  these  calculations  fulfilled,  that 
astronomers  universally  attached  the  name  of  'Encke'  to  the 
comet  of  1819,  not  only  as  an  acknowledgment  of  his  diligence 
and  success  in  the  performance  of  some  of  the  most  intricate 
and  laborious  computations  that  occur  in  practical  astronomy, 
but  also  to  mark  the  epoch  of  the  first  detection  of  a  comet  of 
short  period ;  one  of  no  ordinary  importance  in  this  depart- 
ment of  science." 


ENCKE'S  COMET  219 

The  comet  has  since  been  observed  at  every  reappearance, 
the  appearance  in  1917-1918  being  the  thirty-fourth  on 
record.  In  1835  it  passed  so  near  to  the  planet  Mercury  as 
to  show  conclusively  that  the  generally  received  value  of  that 
planet's  mass  must  be  far  too  great,  since  the  planet  exerted 
no  perceptible  influence  on  the  comet's  orbit. 

The  comet  is  sometimes  visible  to  the  naked  eye.  It 
usually  appears  to  have  no  tail ;  but  in  1848  it  had  two,  one 
about  1°  in  length,  turned  from  the  sun,  and  the  other  of  a 
less  length  and  turned  towards  it.  At  perihelion  the  comet 
passes  within  the  orbit  of  Mercury:  while  at  aphelion  its 
distance  from  the  sun  is  nearly  equal  to  that  of  Jupiter. 

One  very  curious  feature  in  connection  with  this  comet  is 
that  its  period  is  steadily  diminishing,  by  an  amount  of  about 
1\  hours  in  every  revolution,  the  period  having  been  nearly 
1213  days  in  1789-92,  and  only  about  1210  days  in  1862-65. 
Encke's  own  theory  to  account  for  this  diminution  is  that 
the  space  through  which  the  comet  moves  is  filled  with  some 
extremely  rare  medium,  too  rare  to  obstruct  the  motions  of 
the  planets,  but  dense  enough  to  offer  sensible  resistance  to 
the  progress  of  the  comets.  The  effect  of  this  diminution  of 
velocity  is  to  diminish  the  comet's  centrifugal  force,  so  that 
the  comet  is  drawn  nearer  to  the  sun,  and  its  orbit  becomes 
smaller.  But  as  the  orbit  becomes  less,  the  angular  velocity 
of  the  comet  is  increased,  and  its  period  of  revolution  is 
decreased. 

There  has  always  been  more  or  less  doubt  as  to  the  cause 
of  the  retardation  of  Encke's  comet  and  it  seems  quite 
possible  that  this  retardation  may  be  due  to  some  regularly 
recurring  encounter  of  the  comet  with  a  cloud  of  meteoric 
matter.  The  resisting  medium,  whatever  it  may  be,  through 
which  the  comet  passes  is  confined  to  the  orbit  of  this  comet, 
as  no  other  comet  seems  to  be  retarded  by  it. 

Recent  observations  have  established  the  periodicity  of 
the  following  comets : 

(1)  Tempel  (i).  This  comet  was  discovered  by  Tempel,  at 


220  COMETS  AND  METEORIC  BODIES 

Marseilles,  on  May  23,  1867.     It  returned  6.5  years  later 
and  again  in  1879,  but  has  not  been  seen  since  then. 

(2)  Tempel  (ii).    This  comet  was  discovered  by  Tempel, 
July  3,  1873,  having  passed  its  perihelion  June  25. 

It  passed  perihelion  the  second  time,  Sept.  7,  1878,  and 
again  on  Nov.  20,  1883.  Its  latest  observed  perihelion  pass- 
age was  on  Apr.  13,  1915. 

(3)  Ternpel-Swift.     This   comet  appeared  in   1880  and 
was  then  discovered  by  Swift  and  was  identified  with  the 
comet  1869  (iii)  which  had  been  discovered  by  Tempel.     Its 
period  is  about  5J  years. 

(4)  Pons.    The  only  comet  appearing  in  1812  was  dis- 
covered by  Pons  and  is  sometimes  called  the  Comet  of  1812. 
It  was  visible  to  the  naked  eye  and  had  a  tail  2°  in  length. 
The  period  was  computed  to  be  71  years  and  it  reappeared 
in  January,  1884. 

WINNECKE'S  OR  PONS'S  COMET 

245.  The   fifth    comet    in    the    list    was    discovered  by 
Pons,  on  June  12,  1819.     Professor  Encke  assigned  to  it  a 
period  of  5J  years,  but  the  comet  was  not  seen  again  until 
March  8,  1858,  when  it  was  detected  by  Winnecke,  at  Bonn. 
He  was  at  first  inclined  to  consider  it  a  new  comet,  but  soon 
identified  it  with  the  one  previously  discovered  by  Pons. 
Its  distance  from  the  sun  at  perihelion  is  about  70,000,000 
miles,  and  its  distance  at  aphelion  520,000,000  miles.     It 
appeared  in  1869,  1875,  1886,  1898,  1909  and  1915. 

BRORSEN'S  COMET 

246.  This  comet  was  discovered  by  M.  Brorsen,  at  Kiel, 
on  February  26,  1846.     The  orbit  was  found  to  be  elliptical, 
with  a  period  of  about  5J  years,  and  its  return  to  perihelion 
was  fixed  for  September,  1851;  but  its  position^  at  that  time 
was  so  unfavorable  for  observation  that  it  was  not  detected. 


BIELA'S  COMET  221 

It  was  seen  at  its  next  return  to  perihelion,  on  March  29, 
1857.  It  again  escaped  detection  in  1862,  but  was  seen  in 
this  country  on  May  11,  1868.  It  was  seen  also  in  1873, 
1879  and  1890. 

Its  perihelion  distance  is  60,000,000  miles,  and  its  aphelion 
distance,  530,000,000  miles. 

BIELA'S  COMET 

247.  This  comet  was  discovered  by  M.  Biela,  an  Austrian 
officer,  at  Josephstadt,  Bohemia,  on  February  27,  1826.  It 
was  observed  for  nearly  two  months,  and  was  identified  with 
comets  which  had  previously  been  seen  in  1772  and  1805. 

Its  next  return  to  perihelion  was  fixed  for  November  27, 
1832;  and  the  comet  passed  perihelion  within  twelve  hours 
of  that  time.  On  October  29,  1832,  it  passed  within  20,000 
miles  of  the  earth's  orbit:  but  the  earth  did  not  reach  that 
point  of  its  orbit  until  a  month  afterwards.  No  little  alarm 
was  created,  however,  outside  of  the  scientific  world,  when  it 
becanie  generally  known  how  near  to  the  earth's  orbit  the 
comet  would  approach. 

At  its  return  in  1839  it  was  not  observed,  owing  to  its 
close  proximity  to  the  sun.  It  was  again  detected  on  No- 
vember 28,  1845,  and  by  the  end  of  the  year  it  was  found  to 
have  separated  into  two  parts,  and  to  present  the  extraor- 
dinary appearance  of  two  comets,  moving  side  by  side,  at  a 
distance  apart  of  over  150,000  miles.  It  again  returned  in 
1852,  and  presented  the  same  appearance;  but  the  distance 
between  the  parts  had  increased  to  over  one  million  miles. 
Since  that  time  the  comet  has  never  been  seen,  unless  as 
meteroric  showers  in  1872  and  1885. 

Two  theories  have  been  advanced  to  account  for  this 
singular  separation.  One  is  that  the  division  may  have 
been  the  result  of  some  internal  repulsive  force,  similar  to 
that  which  forms  the  tails  of  comets ;  the  other  is  that  it  may 
have  been  the  result  of  collision  with  some  asteroid.  At 


222  COMETS  AND  METEORIC  BODIES 

perihelion  the  comet  passed  within  the  orbit  of  the  earth,  and 
at  aphelion  it  passed  beyond  that  of  Jupiter. 

D'ARREST'S  COMET 

248.  This  comet  was  discovered  by  Dr.   D' Arrest,   at 
Leipsic,  on  June  27,  1851.     It  remained  in  sight  for  about 
three  months,  and  its  period  was  determined  to  be  about 
6^  years.     Its  return  in  November,  1857,  was  accordingly 
predicted,  and  the  prediction  was  verified;  although,  owing  to 
the  comet's  great  southern  declination,  it  was  only  observed 
at  the  Cape  of  Good  Hope.     The  unfavorable  situation  of 
the  comet  in  1864  prevented  its  being  seen ;  but  it  was  seen  in 
1870,  1877,  1890,  1897  and  1910.     It  must  have  passed  its 
perihelion  in  April,  1917,  but  was  not  seen. 

Its  perihelion  distance  is  about  100,000,000  miles,  and 
its  aphelion  distance  more  than  500,000,000  miles. 

FATE'S  COMET 

249.  This  comet  was  discovered  by  M.  Faye,  at  the 
Paris  Observatory,  on  November  22,  1843.     It  had  a  bright 
nucleus  and  a  short  tail,  but  was  not  visible  to  the  naked  eye. 
The  elements  of  its  orbit  were  investigated  by  Leverrier,  who 
predicted  that  it  would  return  to  perihelion  on  April  3,  1851; 
and  it  returned  within  about  a  day  of  the  time  predicted.     It 
has  since  made  nine  returns, — the  last  one  being  in  1910. 
The  dimensions  of  its  orbit  are  nearly  the  same  as  those  of 
D' Arrest's  comet. 

MECHAIN'S  OR  TUTTLE'S  COMET 

250.  This  comet  was  discovered  by  Mechain,  at  Paris,  on 
January  9,  1790.     Its  period  was  calculated  to  be  less  than 
14  years;  but  the  comet  was  not  seen  again  until  January  4, 
1858,  when  it  was  detected  by  Mr.  H.  P.  Tuttle,  at  the 
Harvard  College  Observatory.     Its  sixth  appearance  was  in 
1912. 


HALLEY'S  COMET  223 

HALLEY'S  COMET 

251.  In  the  latter  part  of  the  seventeenth  century  Sir 
Isaac  Newton  published  his  Principia.  In  that  great  work 
he  assumed  that  the  comets  were  analogous  to  the  planets  in 
their  revolutions  about  the  sun,  although  no  periodic  comet 
had  then  been  discovered.  He  explained  the  methods  of 
investigating  the  orbits  of  the  comets,  and  invited  astrono- 
mers to  apply  these  methods  to  the  various  comets  which  had 
been  observed.  Halley,  a  young  English  astronomer,  and 
afterwards  the  second  Astronomer  Royal,  after  a  careful 
investigation,  identified  the  comet  of  1682  with  comets  which 
had  appeared  in  1531  and  1607:  the  period  of  the  comet  being 
about  76  years.  The  fact  that  the  interval  of  time  between 
the  first  and  the  second  of  these  appearances  was  not  exactly 
equal  to  that  between  the  second  and  the  third  seemed  at 
first  to  offer  some  difficulty;  but  Halley,  "with  a  degree  of 
sagacity  which,  considering  the  state  of  knowledge  at  the 
time,  cannot  fail  to  excite  unqualified  admiration,"  advanced 
the  theory  that  the  attractions  of  the  planets  would  exert 
some  influence  on  the  orbits  of  the  comets.  Having  thus 
decided  that  this  comet  was  a  periodic  comet,  Halley  pre- 
dicted the  return  of  the  comet  about  the  beginning  of  the 
year  1759;  and  the  comet  passed  its  perihelion  on  March  12, 
in  that  year.  The  comet  again  appeared  in  1835,  and  its 
next  appearance  was  in  1910. 

The  comet  is  a  very  conspicuous  one,  with  a  tail  sometimes 
30°  in  length  and  sometimes  50°.  The  comet  has  been  traced 
back  through  the  astronomical  records,  with  more  or  less  cer- 
tainty, to  11  B.C.,  the  number  of  appearances  being  about 
eighteen.  It  is  not  impossible  that  it  was  this  comet  which 
appeared  in  1066,  when  it  is  recorded  that  a  large  comet 
excited  dread  throughout  Europe,  and  was  in  England 
considered  to  presage  the  success  of  the  Norman  invasion. 
It  is  also  probably  identical  with  the  comet  of  1456,  which 
had  a  splendid  tail  60°  in  length. 


224  COMETS  AND  METEORIC  BODIES 

Halley's  comet  at  perihelion  is  nearer  to  the  sun  than 
Venus,  while  at  aphelion  it  recedes  beyond  the  orbit  of 
Neptune. 

REMARKABLE  COMETS  OF  THE  NINETEENTH  CENTURY 
THE  GREAT  COMET  OF  1811 

252.  The  comet  of  1811  (i)  was  discovered  on  March 
26,  1811,  and  was  visible  about  seventeen  months.     It  was 
very  conspicuous  in  the  autumn  of  1811,  remaining  visible 
throughout  the  night  for  several  weeks.     Sir  William  Her- 
schel  states  that  the  nucleus  was  well  denned,  with  a  diameter 
of  about  428  miles;  that  it  was  of  a  ruddy  hue,  while  the 
surrounding  nebulous  matter  had  a  bluish-gre'en  tinge.     The 
tail  was  about  25°  in  length,  and  6°  in  breadth.     Its  aphelion 
distance  from  the  sun  is  14  times  that  of  Neptune,  and  its 
period,   according  to  Argelander,  is  3065  years,  with  an 
uncertainty  of  43  years. 

THE  GREAT  COMET  OF  1843 

253.  The  comet  of  1843  (i)  was  first  seen  in  the  southern 
hemisphere  in  February,  and  became  visible  in  the  northern 
hemisphere  the  next  month.     It  was  decidedly  the  most 
wonderful  comet  of  the  last  century.     Its  nucleus  and  coma 
shone  with  great  splendor,  and  its  tail  was  a  luminous  train 
of  about  60°  in  length.     On  the  day  after  its  perihelion 
passage,  and  when  only  4°  distant  from  the  sun,  it  was  seen 
in  broad  daylight  in  some  parts  of  New  England,  and  its 
distance  from  the  sun  was  measured  with  a  sextant.     It  is 
described  as  having  been  at  that  time  as  well  defined,  in 
both  nucleus  and  tail,  as  the  moon  is  on  a  clear  day.     The 
comet  is  remarkable  for  its  small  perihelion  distance,  which 
was  only  about  540,000  miles;  so  that  the  comet  came  within 
100,000  miles  of  the  sun's  surface.     The  intensity  of  the  heat 
to  which  the  comet  must  then  have  been  subjected  is  almost 
inconceivable.     Since  540,000  miles  is  about  TTo^h  of  the 


REMARKABLE  COMETS  OF  NINETEENTH  CENTURY    225 

distance  of  the  earth  from  the  sun,  and  the  intensity  of  heat 
varies  inversely  as  the  square  of  the  distance,  the  heat  to 
which  the  comet  was  subjected  must  have  been  about 
29,000  times  as  intense  as  the  heat  which  prevails  at  the 
earth's  surface:  a  heat  nearly  twenty  times  that  required,  as 
shown  by  experiments  with  powerful  lenses,  to  melt  agate  and 
carnelian.  For  some  days  after  this,  the  tail  had  a  fiery  red 
appearance;  and  its  enormous  length  .of  over  200,000,000 
miles,  and  the  marvelous  rapidity  with  which  it  was  formed, 
were  undoubtedly  the  results  of  the  heat  which  it  endured. 

The  rapidity  with  which  this  comet  moved  about  the  sun 
has  already  been  noticed  (Art.  240).  The  period  has  been 
computed  to  be  about  175  years. 

DONATI'S  COMET 

254.  This  comet,  1858  (vi),  was  discovered  on  June  2d, 
by  Dr.  Donati,  at  Florence.  It  was  then  only  discernible 
with  a  telescope,  but  became  visible  to  the  naked  eye  about 
the  last  of  August.  Indications  of  a  tail  began  to  be  noticed 
about  the  20th  of  August,  and  in  a  few  weeks  the  tail  assumed 
a  noticeable  curvature,  which  subsequently  became  one  of 
the  most  interesting  points  connected  with  the  comet.  The 
comet  passed  its  perihelion  on  September  29th,  and  was  at 
its  least  distance  from  the  earth  on  October  10th.  Its  tail 
subtended  an  angle  of  60°,  and  had  an  absolute  length  of 
51,000,000  miles.  It  disappeared  from  view  in  the  northern 
hemisphere  in  October,  but  was  seen  in  the  southern  hemi- 
sphere until  March,  1859. 

This  comet  was  not  as  large  as  some  others  of  the  comets, 
but  it  was  particularly  noted  for  the  intense  brilliancy  of  its 
nucleus.  The  nebulosity 'surrounding  the  nucleus  was  also 
peculiar  in  its  appearance.  It  consisted  of  seven  luminous 
envelopes,  parabolic  in  form,  and  separated  from  each  other 
by  spaces  comparatively  dark.  These  envelopes  were 
detached  in  succession  from  the  comet's  nucleus,  at  intervals 


226  COMETS  AND  METEORIC  BODIES 

of  from  four  to  seven  days.  They  receded  from  the  nucleus 
with  the  daily  rate  of  about  1000  miles.  Perfectly  straight 
rays  of  light,  or  "secondary  tails/'  were  also  seen. 

The  comet  has  a  period  of  about  2000  years.  A  magnifi- 
cent memoir  of  this  comet,  by  Professor  G.  P.  Bond,  is 
contained  in  the  second  volume  of  the  Annals  of  the  Harvard 
Observatory. 

THE  GREAT  COMET  OF  1861 

255.  This  comet,  the  second  of  the  year,  was  discovered  in 
the  southern  hemisphere  on  May  13th,  but  was  not  seen  in 
England  until  June  29th,  about  two  weeks  after  its  perihelion 
passage.  The  nucleus  was  round  and  unusually  bright,  and 
the  tail  at  one  time  attained  the  length  of  over  100°.  The 
comet  remained  in  sight  for  about  a  year. 

"In  a  letter  published  at  the  tune  in  one  of  the  London 
papers,  Mr.  Hind,  an  English  astronomer,  stated  that. he 
thought  it  not  only  possible,  but  even  probable,  that  in  the 
course  of  Sunday,  June  30th,  the  earth  passed  through  the 
tail  of  the  comet,  at  a  distance  of  perhaps  two-thirds  of  its 
length  from  the  nucleus."  Mr.  Hind  also  stated  that  on 
Sunday  evening  there  was  noticed,  by  both  himself  and  others 
a  peculiar  illumination  in  the  sky,  like  an  auroral  glare;  and  a 
similar  phenomenon  seems  to  have  been  noticed  outside  of 
London. 

According  to  the  observations  of  Father  Secchi,  the  light 
of  the  tail,  and  that  of  the  rays  near  the  nucleus,  presented 
evidences  of  polarization,  while  the  nucleus  itself  at  first 
presented  no  evidences  whatever;  afterwards,  however,  the 
nucleus  presented  decided  indications  of  polarization. 
Secchi  states  that  he  thinks  this  "a  fact  of  great  importance, 
as  it  seems  that  the  nucleus  on  the  former  days  shone  by  its 
own  light,  perhaps  by  reason  of  the  incandescence  to  which  it 
had  been  brought  by  its  close  proximity  to  the  sun." 


REMARKABLE  COMETS  OF  NINETEENTH  CENTURY   227 

THE  GREAT  COMET  OF  1882 

This  comet  was  first  seen  as  a  naked-eye  object  at  Auck- 
land, New  Zealand,  on  September  3d,  but  was  not  seen  in  the 
Northern  hemisphere  until  it  passed  its  perihelion  on  Sep- 
tember 17.  At  this  time  it  was  so  bright  that  it  could  readily 
be  seen  with  the  naked  eye  during  daylight. 

The  afternoon  of  the  17th  the  comet  which  seemed  to  be 
as  bright  as  the  sun's  surface  was  seen  to  approach  the  sun's 
disc  and  was  followed  right  up  to  the  limb  when  it  disappeared 
completely.  It  passed  directly  across  the  disc  but  was  abso- 
lutely invisible;  there  was  not  the  least  trace  of  it  on  the  sun 
so  it  must  have  been  sensibly  transparent.  For  four  days 
after  its  perihelion  it  was  visible  to  the  eye  by  daylight  and  in 
a  few  more  days  it  had  traveled  so  far  from  the  sun  that  it 
was  a  brilliant  feature  of  the  early  morning  sky.  This 
comet  continued  visible  until  March,  so  that  computers  were 
able  to  determine  its  orbit  with  considerable  accuracy;  its 
perihelion  distance  was  less  than  750,000  miles  and  its  orbit 
is  a  very  elongated  ellipse  with  a  period  of  more  than  six 
centuries. 

METEORIC  BODIES 

256.  Under  the  general  head  of  meteors  are  included  three 
classes  of  bodies :  1st.  The  ordinary  shooting  stars,  some  of 
which  can  be  seen  rushing  across  the  heavens  on  almost 
any  clear  night;  2d.  Detonating  meteors,  which  are  shooting 
stars,  commonly  of  an  unusual  size,  whose  disappearance  is 
followed  by  a  sound  like  that  of  an  explosion;  3d.  Aerolites, 
which,  after  the  flash  and  the  explosion  with  which  they  are 
generally  accompanied,  are  precipitated  to  the  earth  in 
showers  of  stones  and  metallic  substances.  It  is  only  within 
recent  years  that  the  decided  attention  of  astronomers  has 
been  directed  to  these  bodies,  and  comparatively  little  is 
known  with  certainty  about  them;  but  the  general  belief  is 
that  they  are  all  essentially  of  the  same  nature,  differing  from 


228 


COMETS  AND  METEORIC  BODIES 


each  other  rather  in  size  and  density  than  in  other  more 
important  respects. 

257.  Shooting  Stars. — Scarcely  a  clear  night  passes  during 
which  shooting  stars  are  not  seen.  The  average  number  of 
those  which  can  be  seen  at  any  place  by  one  observer,  on  a 
cloudless  moonless  night,  •  is  estimated  to  be  about  six  an 
hour.  There  is,  however,  an  hourly  variation  in  the  number 
observed,  the  minimum  occurring  about  6  P.M.,  and  the 
maximum  about  6  A.M.  According  to  a  French  writer  on 
this  subject,  the  mean  number  of  meteors  observed  is  given 
in  the  following  table: 


Hours  

7-8 

9-10 

11-12 

1-2 

3-4 

5-6 

P.M. 

A.M. 

Mean  number.  .  .  . 

3.5 

4 

5 

6.4 

7.8 

8.2 

It  is  further  estimated  that  the  number  seen  at  any  one 
place  by  a  number  of  observers  sufficient  to  watch  the  whole 
hemisphere  of  the  heavens  is  42  an  hour,  on  the  average,  or 
about  1000  daily:  and  that  the  number  which  could  be  seen 
daily  over  the  whole  earth,  under  favorable  circumstances,  is 
more  than  8,000,000.  This  is  the  number  of  those  large 
enough  to  be  visible  to  the  naked  eye:  it  is  simply  impossible 
to  estimate  the  number  of  those  which  could  be  seen  with  the 
aid  of  telescopes. 

It  is  further  noticed  that  there  are  more  shooting  stars 
observed  in  the  second  half  of  the  year  than  in  the  first.  At 
certain  seasons  of  the  year,  either  in  consecutive  years  or 
after  the  lapse  of  a  certain  number  of  years,  there  are  unusu- 
ally brilliant  displays  of  these  meteors,  which  are  called  star 
showers.  The  number  of  recognized  star  showers  now  exceeds 
fifty;  and  prominent  among  them  are  the  shower  of  August 
9--11  and  that  of  November  11-13. 

Professor  Harkness,  of  the  Washington  Observatory, 
after  an  elaborate  investigation  of  the  quantity  of  matter 


METEORIC  BODIES  229 

in  the  ordinary  shooting  star,  concludes  that  it  is  not  far 
from  one  grain. 

258.  The  November  Shower. — There  are  several  histor- 
ical notices  of  brilliant  displays  of  meteors  which  occurred  in 
the  early  centuries  of  the  Christian  era:  and  ten  of  these, 
occurring  between  the  years  902  and  1698,  took  place  in 
October  or  November.  The  first  display,  however,  of  which 
we  have  any  detailed  account,  occurred  in  1799,  on  the 
morning  of  the  13th  of  November,  and  was  visible  over 
nearly  the  whole  of  the  western  continent.  Humboldt  wit- 
nessed it  in  South  America,  and  thus  describes  it:  "Towards 
the  morning  of  the  13th  we  witnessed  a  most  extraordinary- 
scene  of  shooting  meteors.  Thousands  of  bodies  and  falling 
stars  succeeded  each  other  during  four  hours.  Their  direc- 
tion was  very  regular,  from  north  to  south.  From  the  begin- 
ning of  the  phenomenon  there  was  not  a  space  in  the  firma- 
ment equal  in  extent  to  three  diameters  of  the  moon  which 
was  not  filled  every  instant  with  bodies  or  falling  stars.  All 
the  meteors  left  luminous  traces  or  phosphorescent  bands 
behind  them,  which  lasted  seven  or  eight  seconds." 

Similar  showers  also  occurred  on  the  same  day  of  the 
month  in  the  years  1831,  1832,  and  1833,  the  last  one  being 
the  most  splendid  on  record.  It  lasted  from  ten  o'clock  on 
the  night  of  the  12th  to  seven  o'clock  on  the  morning  of  the 
13th,  and  was  visible  over  nearly  the  whole  of  North  America. 
The  display  reached  its  maximum  about  four  A.M.  An 
observer  at  Boston  about  six  o'clock  counted  650  shooting 
stars  in  a  quarter  of  an  hour.  Large  fireballs  with  luminous 
trains  were  also  seen,  some  of  which  remained  visible  for 
several  minutes.  Even  stationary  masses  of  luminous 
matter  are  said  to  have  been  seen:  and  one  in  particular  is 
mentioned  as  having  remained  for  some  time  in  the  zenith 
over  the  Falls  of  Niagara,  emitting  radiant  streams  of  light. 
The  November  shower  was  witnessed  again  in  1866,  both 
in  this  country  and  in  Europe;  but  the  display  was  much 
more  brilliant  in  Europe.  The  maximum  seems  to  have 


230  COMETS  AND  METEORIC  BODIES 

taken  place  about  two  A.M.  on  the  14th,  when  nearly  5000 
meteors  were  counted  in  an  hour  at  Greenwich.  At  half- 
past  one,  124  were  counted  in  one  minute.  The  case  was 
reversed  with  the  shower  of  1867,  the  display  being  more 
brilliant  in  this  country  than  in  Europe.  The  report  on  the 
shower  from  the  United  States  Observatory  at  Washington 
states  that  as  many  as  3000  were  counted  in  one  hour.  The 
most  magnificent  phase  seems  to  have  occurred  about  half- 
past  four  A.M.  on  the  14th.  Professor  Loomis  states  that 
at  New  Haven  about  220  a  minute  were  counted  at  this  time. 
Many  others  were  undoubtedly  rendered  invisible  by  the 
light  of  the  moon,  which  was  then  very  nearly  full.  Most  of 
the  brighter  meteors  left  trains  of  phosphorescent  light, 
which  remained  visible  for  several  seconds,  and  in  some  cases 
for  several  minutes. 

In  1868  the  display  began  somewhat  before  midnight  on 
the  13th  and  continued  until  daybreak  on  the  14th.  Pro- 
fessor Eastman,  of  the  Washington  Observatory,  says  in  his 
report,  that  "considering  the  number  and  brilliancy  of  the 
meteors,  their  magnificent  trains,  and  the  magnitude  of  the 
meteoric  group  through  which  the  earth  passed,  this  shower 
was  unquestionably  the  grandest  that  has  ever  been  witnessed 
at  this  Observatory."  Over  5000  meteors  were  counted, 
and  it  was  estimated  that  at  five  A.M.  on  the  14th  the 
number  falling  in  the  whole  heavens  was  about  2500  an  hour. 
Several  very  brilliant  meteors  were  observed.  One  in 
particular  was  brighter  than  Jupiter.  It  was  at  first  of  a 
deep  orange  color,  afterwards  green  and  finally  light  blue. 
It  left  a  train  of  7°  in  length,  which  passed  through  the  same 
changes  of  color,  and  remained  visible  for  half  an  hour.  The 
paths  of  90  meteors  were  traced  upon  a  chart,  and  were 
found  in  nearly  every  instance  to  start  from  a  point  in  the 
constellation  Leo. 

Extensive  preparations  were  made  around  the  world  in 
1898,  1899,  and  1900,  for  observing  the  fall  of  meteors  in 
November,  and  the  number  observed  was  decidedly  greater 
than  it  commonly  is  at  that  period  of  the  year.  At  the 


METEORIC  BODIES  231 

Harvard  Observatory,  for  instance,  on  November  14,  1898, 
800  meteors  were  observed,  and  227  trails  were  charted. 
But  nowhere  was  there  a  shower  of  meteors  like  the  showers 
reported  in  1868. 

259.  Height,  etc.,  of  the  Meteors. — Concurrent  observa- 
tions were  made  at  Washington  and  Richmond,  in  November, 
1867,  for  the  purpose  of  determining  the  parallax  of  the  me- 
teors, and  thence  their  distance.     It  was  found  that  they 
appeared  at  an  average  height  of  75  miles,  and  disappeared 
at  the  height  of  55  miles.     The  velocity  with  which  they 
moved  relatively  to  the  earth  was  44  miles  a  second.     Other 
observations  have  given  nearly  the  same  results. 

The  light  of  the  meteors  is  probably  due  to  the  intense 
heat  generated  by  the  resistance  of  the  air  to  the  progress 
of  these  bodies.  Notwithstanding  the  extreme  rarity  of  the 
air  at  the  height  of  the  meteors,  it  is  still  believed  that  the 
heat  resulting  from  such  immense  velocity  is  sufficient  to 
fuse  any  known  substance.  A  body  moving  with  this 
velocity  at  the  earth's  surface  would  acquire  a  temperature 
of  at  least  3,000,000°.  An  examination  of  the  light  of  the 
meteors  with  the  aid  of  the  spectroscope,  by  Mr.  A.  Herschel, 
showed  that  some  of  the  meteors  were  solid  bodies  in  a  state 
of  ignition,  but  that  most  of  them  were  gaseous. 

260.  Orbits  of  the  Meteors. — It  is  noticed  that  the  No- 
vember meteors,  or  at  all  events  the  great  majority  of  them, 
seem  to  come  from  the  same  point  in  the  heavens, — a  point 
in  the  constellation  Leo.     So  also  the  August  meteors  come 
from  a  point  near  the  head  of  the  constellation  Perseus. 
Such  points  are  called  radiant  points.     Other  showers  have 
also  other  radiant  points,  situated  in  various  parts  of  the 
heavens.     The  number  of  such  points  now  recognized  is  more 
than  60.     The  paths  in  which  meteors  having  the  same 
radiant  point  move  during  the  instant  of  time  that  we  see 
them,  are  really  parallel  straight  lines,  the  apparent  conver- 
gence of  the  paths  being*  merely  the  result  of  perspective; 
in  other  words,  the  radiant  point  is  the  vanishing  point 
(Art.  16)  of  these  parallel  lines. 


232  COMETS  AND  METEORIC  BODIES 

Knowing  the  direction  and  the  velocity  with  respect  to 
the  earth  of  the  motion  of  a  meteor,  it  is  easy  to  compute 
the  same  elements  of  its  motion  with  reference  to  the  sun. 
The  results  of  such  computation  together  with  the  existence 
of  the  radiant  points  and  the  periodic  recurrence  of  showers, 
have  led  to  the  theory  that  the  November  meteors  are 
collected  in  a  ring,  or  in  several  rings,  or  possibly  in  a  series  of 
clusters  or  groups,  which  revolve  about  the  sun;  and  that  the 
showers  occur  when  the  earth  encounters  these  rings  or 
groups.  The  following  account  of  the  way  in  which  this 
theory  forced  itself  upon  the  attention  of  scientific  men  is 
taken  from  Clerke's  History  of  Astronomy  in  the  Nineteenth 
Century : 

"Once  for  all,  then,  as  the  result  of  the  star-fall  of  1833, 
the  study  of  luminous  meteors  became  an  integral  part  of 
astronomy.  .  .  .  Evidences  of  periodicity  continued  to 
accumulate.  It  was  remembered  that  Humboldt  and 
Bonpland  had  been  the  spectators  at  Cumana,  after  mid- 
night of  November  12,  1799,  of  a  fiery  shower  little  inferior 
to  that  of  1833,  and  reported  to  have  been  visible  from  the 
equator  to  Greenland.  Moreover,  in  1834,  and  some  subse- 
quent years,  there  were  waning  repetitions  of  the  display,  as 
if  through  the  gradual  thinning  out  of  the  meteoric  supply. 
The  extreme  irregularity  of  its  distribution  was  noted  by 
Olbers  in  1837,  who  conjectured  that  we  might  have  to  wait 
until  1867  to  see  the  phenomenon  renewed  on  its  former 
scale  of  magnificence.  This  was  the  first  hint  of  a  33-  or  34- 
year  period. 

"The  falling  stars  of  November  did  not  alone  attract  the 
attention  of  the  learned.  Similar  appearances  were  tradi- 
tionally associated  with  August  10  by  the  popular  phrase  in 
which  they  figured  as  'the  tears  of  St.  Lawrence.'  But  the 
association  could  not  be  taken  on  trust  from  mediaeval  au- 
thority. It  had  to  be  proved  scientifically,  and  this  Quetelet 
of  Brussels  succeeded  in  doing  in  December,  1836." 

By  noting  the  duration  of  a  shower,  and  combining  it 
with  the  velocity  of  the  earth  in  its  orbit,  we  can  obtain  an 


METEORIC  BODIES 


233 


approximate  value  of  the  breadth  of  the  ring.  Professor 
Eastman  estimates  that  the  breadth  of  that  portion  of  the 
ring  through  which  the  earth  passed  in  November,  1868, 
could  not  have  been  less  than  115,000  miles.  The  breadth 
of  the  stream  in  1867  was  less  than  this,  but  more  densely 
packed  with  meteors. 

Leverrier,  a  French  astronomer,  has  computed  the 
elements  of  the  orbit  of  the  November  meteors.  He  finds  the 
major  semi-axis  to  be  10.34, 
the  perihelion  distance  0.989 
(the  radius  of  the  earth's 
orbit  being  unity),  and  the 
eccentricity  0.9044.  This 
would  carry  the  aphelion 
beyond  the  orbit  of  Uranus, 
if  both  orbits  were  projected 
upon  the  plane  of  the  eclip- 
tic. 

According  to  Professor 
Loomis,  the  relative  situa- 
tions of  the  orbit  of  the 
November  meteors  and  the 
orbits  of  the  earth  and  the 
other  planets,  are  repre- 
sented in  Fig.  74,  the  orbit 
of  the  meteors  being  a  very 
eccentric  ellipse,  the  aphe- 
lion of  which  lies  beyond 
the  orbit  of  Uranus,  and  the 
period  of  the  meteors  being 
33J  years.  According  to  JPIG  74 

the    same    authority,    the 

August  meteors  revolve  in  a  similar  but  much  more  eccentric 
ellipse,  of  which  the  aphelion  lies  far  beyond  the  orbit  of 
Neptune.  The  theory  has  also  been  advanced  that  meteors 
(or,  at  all  events,  some  of  them)  are  to  be  regarded 
as  satellites  of  the  earth  rather  than  of  the  sun.  On  this 


234  COMETS  AND  METEORIC  BODIES 

subject  Sir  John  Herschel  says,  in  his  Outlines  of  Astronomy: 
"It  is  by  no  means  inconceivable  that  the  earth,  approaching 
to  such  as  differ  but  little  from  it  in  direction  and  velocity,  may 
have  attached  them  to  it  as  permanent  satellites,  and  of  these 
there  may  be  some  so  large  as  to  shine  by  reflected  light,  and 
to  become  visible  for  a  brief  moment;  suffering,  after  that, 
extinction  by  plunging  into  the  earth's  shadow.'7 

261.  Detonating  Meteors. — The  height  and  the  velocity 
of  these  bodies  are  not  essentially  different  from  those  of  the 
ordinary  shooting  stars.  They  are,  however,  generally 
of  an  unusual  brilliancy,  and  their  appearance  is  followed  by 
an  explosion,  or  a  series  of  explosions,  the  intensity  of  which 
is  sometimes  terrific.  Records  of  more  than  eight  hundred 
detonating  meteors  are  to  be  found  in  scientific  journals. 
The  phenomena  connected  with  the  appearance  of  these 
bodies  are,  however,  so  nearly  identical  in  character,  that  one 
instance  may  suffice  to  exemplify  all.  "On  the  2d  of  August 
1860,  about  10  P.M.;  a  magnificent  fireball  was  seen  through- 
out the  whole  region  from  Pittsburg  to  New  Orleans,  and 
from  Charleston  to  St.  Louis,  an  area  of  900  miles  in  diameter. 
Several  observers  described  it  as  equal  in  size  to  the  full  moon, 
and  just  before  its  disappearance  it  broke  into  several  frag- 
ments. A  few  minutes  after  the  flash  of  the  meteor  there 
was  heard  throughout  several  counties  of  Kentucky  and 
Tennessee  a  tremendous  explosion,  like  the  sound  of  distant 
cannon.  Immediately  another  noise  was  heard,  not  quite 
so  loud,  and  the  sounds  were  re-echoed  with  the  prolonged 
roar  of  thunder.  From  a  comparison  of  a  large  number  of 
observations,  it  has  been  computed  that  this  meteor  first 
became  visible  over  Northeastern  Georgia,  about  82  miles 
above  the  earth's  surface,  and  that  it  exploded  over  the 
southern  boundary  line  of  Kentucky,  at  an  elevation  of  28 
miles.  The  length  of  its  visible  path  was  about  240  miles, 
and  its  time  of  flight  eight  seconds :  showing  a  velocity  relative 
to  the  earth  of  30  miles  per  second.  It  is  hence  computed 
that  its  velocity  relative  to  the  sun  was  24  miles  per  second." 


METEORIC  BODIES  235 

The  explosions  are  probably  due  to  the  sudden  compres- 
sion and  shocks  to  which  the  air  is  subjected  as  the  meteor 
rushes  through  it,  as  happens  when  a  gun  is  fired;  or  to  the 
rushing  of  the  air  into  the  vacuum  which  the  body  creates  in 
its  rear.  The  appearance  of  these  bodies  is  so  sudden,  and 
their  velocity  so  great,  that  it  is  almost  impossible  to  obtain 
any  definite  value  of  their  magnitude.  The  diameters 
of  some  of  them  are  stated  to  have  been  several  thousand 
feet  in  length,  but  the  estimate  must  be  taken  with  consider- 
able caution,  particularly  as  it  is  impracticable  to  distinguish 
between  the  meteor  itself  and  the  blaze  of  light  which  sur- 
rounds it. 

262.  Aerolites. — Although  the  ordinary  shooting  stars 
sometimes  appear  to  break  in  pieces,  there  is  no  evidence 
that  any  part  of  them  falls  to  the  earth.  But  occasionally 
solid  masses  of  stone  or  of  metallic  substances  do  fall  to  the 
earth,  their  fall  being  usually  preceded  by  the  flash  and  the 
discharge  of  a  detonating  meteor.  There  is  no  doubt  what- 
ever about  the  authenticity  of  most  of  these  cases  and  the 
record  of  them  extends  far  back  into  ancient  history.  A 
fall  of  meteoric  stones  near  Rome,  650  years  before  Christ,  is 
mentioned  by  the  historian  Livy ;  and  a  large  block  of  stone 
is  said  to  have  fallen  in  Thrace,  near  what  is  now  called  the 
Strait  of  Dardanelles,  465  years  before  Christ.  The  entire 
number  of  aerolites  of  which  we  have  any  determinate 
knowledge  is  more  than  400;  and  more  than  twenty  falls  of 
aerolites  occurred  in  the  United  States  during  the  nineteenth 
century.  The  British  Museum  contains  a  large  collection  of 
aerolites,  one  of  which  weighs  8287  pounds ;  and  many  .other 
similar  specimens  are  to  be  found  in  the  cabinets  of  colleges 
and  museums,  both  in  this  country  and  in  Europe.  The 
following  are  instances  of  falls  which  have  occurred  since  1800. 

In  1807,  on  the  morning  of  December  14th,  a  brilliant 
meteor,  with  an  apparent  diameter  equal  to  about  one-half 
of  that  of  the  moon,  was  seen  moving  over  the  town  of 
Weston,  in  the  southwestern  part  of  Connecticut.  After  its 


236  COMETS  AND  METEORIC  BODIES 

disappearance,  three  loud  explosions  were  heard,  followed 
by  a  continuous  rumbling.  Fragments  of  stone  were  pre- 
cipitated to  the  earth  within  an  area  of  a  few  miles  in  diam- 
eter. The  entire  weight  of  these  fragments  is  estimated  to 
be  about  300  pounds.  One  fragment,  weighing  36  pounds, 
is  preserved  in  the  museum  at  Yale  College.  The  specific 
gravity  of  the  aerolite  was  about  3J,  and  among  its  com- 
ponents were  silex,  oxide  of  iron,  magnesia,  nickel,  and  sul- 
phur. 

On  the  1st  of  May,  1860,  about  noon,  there  were  a 
number  of  explosions  over  the  southeastern  part  of  Ohio. 
Stones  were  seen  to  fall  to  the  earth,  and  in  some  cases  they 
penetrated  the  earth  to  a  distance  of  three  feet.  About 
thirty  fragments  were  found,  the  largest  of  which  weighs 
103  pounds,  and  is  to  be  found  in  the  cabinet  of  Marietta 
College.  The  combined  weight  of  these  thirty  fragments  is 
not  far  from  700  pounds,  and  the  specific  gravity  and  the 
composition  are  very  similar  to  those  of  the  Weston  aerolite. 

Another  phenomenon  of  this  character  occurred  in 
Piedmont,  on  February  29,  1868,  about  the  middle  of  the 
forenoon.  There  was  a  heavy  discharge  like  that  of  artillery, 
followed,  after  a  short  interval,  by  a  second  discharge.  A 
mass  of  irregular  shape  was  seen  in  the  air,  enveloped  in 
smoke,  and  followed  by  a  long  train  of  smoke.  Other 
bodies,  similar  in  appearance  to  meteors,  were  also  seen. 
The  analysis  of  the  stones  which  fell  showed  the  existence  in 
them  of  the  components  mentioned  above,  and  also  of  cop- 
per, manganese,  and  potassium. 

Other  aerolites  have  been  subjected  to  chemical  analysis. 
Of  the  70  elementary  substances  known,  24  at  least  have  been 
found  in  aerolites,  and  no  new  elements  have  been  discovered. 
A  meteoric  shower  usually  consists  of  meteoric  iron  and 
meteoric  stone;  the  iron  is  an  alloy  of  which  the  principal 
part  is  nickel,  and  which  also  contains  cobalt,  tin,  copper, 
manganese,  and  carbon;  the  stone  contains  chiefly  those 
minerals  which  are  abundant  in  lava  and  trap-rock.  The 


METEORIC  BODIES  237 

proportions  in  which  these  ingredients  enter  into  the  com- 
position of  different  aerolites  differ  greatly;  sometimes  an 
aerolite  contains  96  per  cent  of  iron,  sometimes  scarcely 
any  iron  at  all.  A  substance  called  schreibersite,  which  is  a 
compound  of  iron,  nickel,  and  phosphorus,  is  always  found  in 
these  bodies. 

The  explosion  of  an  aerolite  may  be  due  either  to  the 
intense  heat  generated  by  its  rapid  motion,  or  to  the  pres- 
sure to  which  it  is  subjected  by  the  resistance  of  the  atmos- 
phere. 

263.  Origin  of  Aerolites* — Many  theories  have  been  ad- 
vanced to  explain  the  origin  of  these  bodies.     One  theory  is 
that  they  may  be  formed  in  the  atmosphere  by  the  aggrega- 
tion of  minute  particles  drawn  up  from  the  surface  of  the 
earth;   another  is  that  they  are  thrown  from  terrestrial 
volcanoes.     One  objection,  among  others,  to  these  theories 
is  that  although,  as  already  stated,  no  new  elements  have 
been  found  in  these  bodies,  the  combinations  of  these  elements 
are  different  from  combinations  found  on  the  earth.     A 
third  theory,  that  they  may  be  ejected  from  the  volcanoes 
of  the  moon,  is  weakened  by  the  fact  that  observation  shows 
no  signs  (or  at  least  almost  no  signs)  of  activity  in  the  lunar 
volcanoes.     The  most  probable   theory   is   that   they   are 
satellites  of  the  sun,  revolving  about  it  in  orbits  which 
intersect  the  orbit  of  the  earth,  and  that  their  fall  to  the 
earth's  surface  either  is  the  direct  result  of  their  own  motion, 
or  is  due  to  the  resistance  of  the  atmosphere  and  the  attrac- 
tion exerted  upon  them  by  the  earth. 

264.  Possible    Connection    of    Comets    and    Meteoric 
Bodies. — The  facts  which  have  been  presented  in  this  chapter 
in  relation  to  comets  and  meteoric  bodies  point  to  one  certain 
conclusion:    that   space,  or  at  least  that  portion  of  space 
through  which  the  earth  moves,  must  be  considered  to  be 
filled  with   a   countless  number  of  comparatively  minute 
bodies,  the  aggregate  mass  of  which  cannot  fail  to  be  very 
#reat.     In  1848  Dr.  Mayer,  of  Germany,  advanced  a  theory 


238  COMETS  AND  METEORIC  BODIES 

that  the  light  and  the  heat  of  the  sun  are  caused  by  the 
incessant  collision  of  meteoric  bodies  with  its  surface.  In 
connection  with  this  subject,  Professor  William  Thompson 
states  that  if  the  earth  were  to  fall  into  the  sun,  the  amount 
of  heat  generated  by  the  shock  would  be  equal  to  that 
which  the  sun  now  gives  out  in  95  years;  and  that  the  planet 
Jupiter,  under  similar  circumstances,  would  generate  an 
amount  of  heat  equal  to  that  given  out  by  the  sun  in 
32,000  years. 

There  is  a  striking  similarity  between  the  elements  of 
the  orbit  of  Temple's  comet,  1866  (i),  and  those  of  the 
November  shower.  There  are  also  at  least  70  other  cases  of 
similar  coincidences,  notably  that  of  the  August  shower  and 
the  Great  Comet  of  1862;  and  the  opinion  is  gaining  ground 
with  astronomers,  not  only  that  each  of  these  comets  leads 
the  group  with  the  elements  of  whose  orbit  its  own  elements 
so  nearly  coincide,  but  also  that  there  is  a  close  connection, 
generally,  between  comets  and  meteoric  bodies.  The  follow- 
ing statement  of  this  theory  is  taken  from  an  article  by 
Professor  Simon  Newcomb,  in  the  North  American  Review 
of  July,  1868. 

"The  planetary  spaces  are  crowded  with  immense  num- 
bers of  bodies  which  move  around  the  sun  in  all  kinds  of 
erratic  orbits,  and  which  are  too  minute  to  be  seen  with  the 
most  powerful  telescopes. 

"If  one  of  these  bodies  is  so  large  and  firm  that  it  passes 
through  the  atmosphere  and  reaches  the  earth  without  being 
dissipated,  we  have  an  aerolite. 

"If  the  body  is  so  small  or  so  fusible  as  to  be  dissipated  in 
the  upper  regions  of  the  atmosphere,  we  have  a  shooting 
star. 

"A  crowd  of  such  bodies  sufficiently  dense  to  be  seen  in 
the  sunlight  constitutes  a  comet. 

"A  group  less  dense  will  be  entirely  invisible,  unless  the 
earth  happens  to  pass  through  it,  when  we  shall  have  a 
meteoric  shower." 


METEORIC  BODIES  239 

It  is  not  impossible  to  conceive  that  the  planets 
themselves  may  have  been  formed  by  the  aggregation 
of  these  minute  bodies,  a  method  of  formation  exactly 
the  opposite  of  that  which  is  set  forth  in  the  nebular 
hypothesis. 


CHAPTER  XIV 

THE  FIXED  STARS.     NEBULA.     MOTION  OF  THE  SOLAR 

SYSTEM 

i 

265.  WE  have  now  examined  the  motions  and  the  orbits 
of  all  the  known  members  of  the  solar  system.  We  have  seen 
that  the  planets  and  their  satellites,  besides  their  apparent 
diurnal  motion  towards  the  west  in  orbits  whose  planes  are 
perpendicular  to  the  axis  of  the  celestial  sphere,  have  also 
independent  motions  of  their  own  in  elliptical  orbits,  the 
sun  being  at  the  common  focus  of  the  orbits  of  the  planets, 
and  each  planet  being  at  the  common  focus  of  the  orbits  of 
its  satellites.  Besides  these  bodies,  there  is  a  vast  number  of 
other  bodies  visible  in  the  heavens,  the  phenomena  presented 
by  which  are  radically  different  from  those  which  have  hith- 
erto been  noticed.  Continuous  observations  have  been  made 
upon  the  stars  from  year  to  year,  and  even  from  century  to 
century;  and  it  has  been  found  that,  after  the  results  of  these 
observations  have  been  freed  from  the  effects  of  precession 
and  nutation  (which,  by  shifting  the  position  of  the  points 
or  the  planes  of  reference,  may  give  the  stars  an  apparent 
motion),  the  real  change  of  position  of  the  stars  is  extremely 
small.  Sirius,  the  brightest  star  in  the  whole  heavens,  has 
an  annual  motion  of  1";  a  Centauri,  the  brightest  star  in 
the  southern  hemisphere,  has  an  annual  motion  of  nearly 
4";  a  star  of  the  seventh  magnitude,  Groombridge  1830,  has 
a  motion  of  7",  and  a  faint  star  observed  by  Barnard  at 
the  Yerkes  Observatory  has  a  proper  motion  of  over  10" 
annually,  the  greatest  yet  observed.  In  only  a  few  hun- 
dred, however,  has  the  amount  of  this  change  of  position 
been  found  to  be  as  great  as  1"  a  year;  and  in  the  others,  it 

240 


MAGNITUDES  OF  THE  FIXED  STARS  241 

is  only  that  of  a  few  seconds  in  a  century.  These  motions 
are  called  proper  motions,  to  distinguish  them  from  those 
which  are  only  apparent,  and  the  stars  are  called  fixed  stars: 
a  term  which  must  be  understood  to  imply,  not  that  they 
have  no  motion  in  space,  but  that  whatever  motion  they  have 
makes  no  perceptible  alteration  in  their  position  upon  the 
celestial  sphere. 

When  a  star  moves  obliquely  to  the  line  joining  the  earth 
and  the  star,  its  motion  in  its  orbit  can  be  resolved  into  two 
motions:  one  along  the  line  of  sight,  either  directly  towards 
the  earth  or  directly  from  it,  and  the  other  at  right  angles  to 
that  line.  This  latter  motion  will  cause  the  star  to  shift  its 
apparent  position  upon  the  celestial  sphere,  and  will  be  the 
proper  motion  above  described;  or,  as  it  may  be  called,  the 
transverse  proper  motion.  Now,  it  is  evident  that  in  order  to 
obtain  the  real  motion  of  any  star  in  space  we  must  be  able 
to  determine,  not  only  its  transverse  motion,  but  also  its 
motion  towards  or  from  the  earth.  As  long  as  the  detection 
of  such  a  motion  depended  upon  our  ability  to  detect  either 
an  increase  or  a  diminution  of  the  star's  brightness,  its 
immense  distance  from  us  rendered  the  task  a  hopeless  one; 
but  very  recently  the  spectroscope  has  afforded  us  the  means 
of  solving  this  problem.  The  details  of  this  method  will  be 
given  at  the  end  cf  this  chapter. 

266.  The  Number  of  the  Fixed  Stars.— The  number  of 
stars  in  the  entire  sphere  which  are  visible  to  the  naked  eye 
in  the  northern  hemisphere  is  between  6000  and  7000;  but 
Dr.   Gould  catalogues  more  than   10,000  stars  which  are 
visible  in  the  clear  atmosphere  of  Cordoba.     By  means  of 
telescopes,  thousands  and  even  millions  of  other  stars  are 
brought  into  view,  in  such  numbers  as  almost  to  defy  any 
attempt  at  computation. 

267.  Magnitudes. — The  fixed  stars  are  classified  arbi- 
trarily by  astronomers  according  to  their  relative  brightness, 
the  different  classes  receiving  the  name  of  magnitudes,  and 
the  first  magnitude  comprising  those  stars  which  are  the 


242  THE  FIXED  STARS.    NEBULA 

brightest.  Different  astronomers,  however,  sometimes 
assign  different  magnitudes  to  the  same  star.  According  to 
Argelander's  classification,  there  are  20  stars  of  the  first 
magnitude,  65  of  the  second,  190  of  the  third,  425  of  the 
fourth,  etc.,  the  numbers  in  the  following  magnitudes  increas- 
ing very  rapidly.  It  is  estimated  that. there  are  at  least 
20,000,000  stars  in  the  first  fourteen  magnitudes.  Those 
stars  which  are  visible  to  the  naked  eye  are  comprised  in  the 
first  six  magnitudes;  and  stars  of  the  twentieth  magnitude 
are  detected  with  the  most  powerful  telescopes. 

(1)  If  the  star  Aldebaran  (a  Tauri)  is  taken  as  the 
standard  of  first  magnitude,  stars  of  the  sixth  magnitude 
are  approximately  of  one  hundredth  the  brightness  of  Alde- 
baran. Using  this  scale  stars  of  any  magnitude  are  approxi- 
mately 2\  times  as  bright  as  stars  of  the  next  lower  magni- 
tude. 

While  stars  of  the  same  magnitude  are,  strictly  speaking, 
of  equal  brightness,  a  less  rigid  definition  of  the  term  mag- 
nitude allows  stars  of  quite  different  degrees  of  brightness 
to  be  classed  as  of  the  same  magnitude.  Using  the  popular 
definition,  stars  brighter  than  a  magnitude  of  1 . 5  are  said 
to  be  of  the  first  magnitude,  those  from  1.5  to  2.5  of  the 
second  magnitude, etc.;  Sirius  (magnitude  —1.6)  andRegulus 
(magnitude  1.3)  are  both,  popularly,  called  stars  of  the  first 
magnitude,  yet  Sirius  is  fifteen  times  as  bright  asRegulus.* 

*  Any  one  of  the  most  prominent  stars  may  be  identified  when  on 
the  meridian,  in  the  following  manner.  The  right  ascension  of  the 
star,  given  in  the  Ephemeris,  is,  by  Art.  9,  the  local  sidereal  time  of 
the  star's  transit;  and  from  this  sidereal  time  the  local  mean  solar 
time  can  be  obtained,  by  subtracting  from  it  the  right  ascension  of 
the  mean  sun.  The  right  ascension  of  the  mean  sun,  to  obtain  exact 
results,  must  be  corrected  for  the  longitude  and  the  sidereal  interval 
must  be  reduced  to  a  mean  time  interval.  (See  example  in  the  latter 
part  of  the  Ephemeris  or  Nautical  Almanac.)  The  data  for  an  approxi- 
mate method  which  will  give  sufficiently  accurate  results  for  most 
purposes  may  be  found  in  the  Nautical  Almanac;  the  Greenwich 
mean  time  of  transit  at  Greenwich  is  tabulated  for  the  first  day  of  each 


REMARKABLE  CONSTELLATIONS  243 

268.  Constellations. — In  order  to  facilitate  the  formation 
of  catalogues  of  the  stars,  they  are  separated  into  groups, 
called    constellations.     Ptolemy,    in    the    second    century, 
enumerated  48  constellations:  21  northern,  12  zodiacal,  and 
15  southern.     The  twelve  zodiacal  constellations  have  the 
same  names  that  the  signs  of  the  zodiac  bear,  which  are  given 
in  Art.  91;  indeed,  the  signs  really  took  their  names  from  the 
constellations.     Owing,  however,  to  the  precession  of  the 
equinoxes,  the  signs  and  the  constellations  no  longer  coincide 
(see  Art.  119),  the  constellation  of  Aries  being  in  the  sign  of 
Taurus,  etc. 

Since  the  time  of  Ptolemy,  about  sixty  other  constella- 
tions have  been  added  to  the  list.  Not  all  of  these,  how- 
ever, are  accepted  by  astronomers,  and  the  list  of  those 
constellations  which  are  generally  acknowledged  com- 
prises only  about  86  :  28  northern,  12  zodiacal,  and  46 
southern. 

269.  Remarkable  Constellations. — The  most  remarkable 
of  the  northern  constellations  is  that  called  Ursa  Major,  or 
the  Great  Bear,  often  called  "The  Dipper"  from  the  well- 
known  appearance  presented  by  its  seven  conspicuous  stars. 
The  two  of  these  seven  stars  which  are  the  most  remote  from 
the  handle  of  the  dipper  are  called  the  pointers,  since  the 
right  line  joining  them,  when  prolonged,  passes  very  nearly 
through   the   pole-star.     This   constellation   contains  fifty- 
three  stars  of  the  first  five  magnitudes,  including  one  of  the 
first  and  three  of  the  second. 

There  is  another  "Dipper,"  much  less  conspicuous,  con- 
sisting also  of  seven  stars,  the  pole-star  being  at  the  extremity 
of  the  handle.  These  stars  form  a  part  of  the  constellation 
Ursa  Minor,  which  contains,  in  all,  twenty-three  stars  of  the 

month  and  an  auxiliary  table  gives  corrections  to  be  applied  to  the 
mean  time  of  transit  on  the  first  day  of  the  month  to  find  the  mean 
time  of  transit  on  any  other  day  of  the  month. 

The  star's  meridian  altitude  is  found  by  the  method  given  in  the 
note  to  Art.  181. 


244  THE  FIXED  STARS.     NEBULAE 

first  five  magnitudes.  The  pole-star  itself  is  of  the  second 
magnitude. 

The  constellation  Orion  is  a  magnificent  one,  and  was 
fancifully  supposed  by  the  ancients  to  bear  some  resemblance 
to  a  giant.  It  contains  two  stars  of  the  first  magnitude,  four 
of  the  second,  and  thirty-one  of  the  next  three  magnitudes. 
The  three  stars,  situated  nearly  in  a  straight  line,  which  form 
the  giant's  belt,  are  a  very  conspicuous  part  of  the  constella- 
tion. This  constellation,  in  the  northern  hemisphere,  bears 
south  about  9  P.M.  in  the  early  part  of  February;  its  altitude 
at  that  time  at  any  place  being  nearly  equal  to  the  co-latitude 
of  that  place.  Sirius,  the  brightest  star  in  the  heavens,  is  also 
seen  at  that  time  to  the  left  of  this  constellation,  and  a  little 
below  it. 

The  constellation  Pegasus  contains  forty-three  stars  of  the 
first  five  magnitudes.  Four  of  these  stars  are  of  the  second 
magnitude,  and  nearly  form  a  square.  This  square  bears 
south  about  9  P.M.  in  the  early  part  of  October,  with  an 
altitude,  in  the  northern  hemisphere,  about  15°  greater  than 
the  co-latitude  of  the  place  of  observation. 

The  constellation  Gemini,  or  the  Twins,  takes  its  name 
from  two  bright  stars,  nearly  of  the  first  magnitude,  called 
Castor  and  Pollux.  They  are  situated  near  each  other,  and 
bear  south  about  9  P.M.  in  the  latter  part  of  February,  in 
the  northern  hemisphere,  their  altitude  being  about  30° 
greater  than  the  co-latitude  of  the  place  of  observation. 

270.  Stars  of  the  Same  Constellation. — Stars  of  the  same 
constellation  are  distinguished  from  each  other  by  the 
letters  of  the  Greek  or  the  Roman  alphabet,  or  by  numerals. 
The  Greek  letters  were  first  used  by  Bayer,  a  German 
astronomer,  in  1604,  who  called  the  brightest  star  in  a  con- 
stellation a,  the  next  brighest  /3,  etc.  Thus  the  pole-star 
bears  the  astronomical  name  of  a  Ursae  Minoris,  and  the 
pointers  of  the  dipper  are  called  a  and  /3  Ursse  Majoris.  Ow- 
ing, however,  either  to  carelessness  on  the  part  of  Bayer  or  to 
changes  in  the  brightness  of  some  of  the  stars,  this  alphabet- 


STARS  WITH  SPECIAL  NAMES 


245 


ical  arrangement  does  not  in  all  cases  accurately  represent 
the  relative  brilliancy  of  the  stars  in  a  constellation. 

The  entire  number  of  stars  now  catalogued  amounts  to 
several  hundreds  of  thousands.  Three  catalogues  published 
by  Argelander,  in  1859-62,  contain  over  320,000  stars 
observed  at  Bonn.  In  large  catalogues,  the  stars  are  usually 
numbered  from  beginning  to  end  in  the  order  of  their  right 
ascensions. 

271.  Stars  with  Special  Names. — Some  of  the  stars, 
particularly  the  more  conspicuous  ones,  have  special  names, 


*a  Eridani Achernar 

a  Tauri Aldebaran 

a  Aurigse Capella 

a  Orionis Betelgeuse 

/3  Orionis Rigel 

*<x  Argus Canopus 

a  Canis  Majoris Sirius 

a  Canis  Minoris Procyon 

ft  Geminorum Pollux 

a  Leonis Regulus 

a  Virginis Spica 

a  Bootis.  .* Arcturus 

a  Scorpii Antares 

a  Lyrae Vega 

a  AquilsB Altair 

a  Piscis  Australis Fomalhaut 

a  Cygni Deneb 


which  were  given  to  them  by  ancient  astronomers.  Instances 
are  given  in  the  accompanying  table,  all  the  stars  contained 
in  it  being  commonly  considered  to  be  of  the  first  magnitude. 
About  90  stars  are  thus  named,  though  most  of  these  names 
except  those  given  in  the  table  are  rarely  used. 

All  of  these  stars,  excepting  those  marked  with  an  asterisk, 
come  above  the  horizon  throughout  the  United  States.  The 
position  of  each,  in  right  ascension  and  declination,  can  be 


246  THE  FIXED  STARS.    NEBULA 

found  in  the  Ephemeris  for  any  day  in  the  year.  Besides 
the  stars  in  the  preceding  table,  there  are  three  others  of  the 
first  magnitude:  a  Crucis,  a  Centauri,  and  0  Centauri.  They 
are  all  stars  of  large  southern  declination,  and  do  not  come 
above  the  horizon  in  any  part  of  the  United  States  excepting 
the  southern  parts  of  Texas  and  Florida. 

272.  Constitution  and  Diversity  of  Brightness. — The 
spectra  of  stars  are  found  to  contain  dark  lines,  often  similar 
in  character  to  those  by  which  we  have  seen  that  the  solar 
spectrum  is  distinguished  (Art.  102).  These  systems  of  lines 
differ  from  the  system  of  lines  in  the  solar  spectrum,  and  they 
are  also  different  in  different  stars.  This  difference  usually 
consists  in  the  absence  of  certain  lines  seen  in  the  solar 
spectrum,  and  not  in  the  presence  of  new  ones:  but  some 
stars  show  lines  which  have  no  counterpart  in  the  sun. 
The  examination  of  these  spectra,  and  the  comparison  of  the 
dark  lines  which  they  contain  with  the  bright  lines  found  in 
the  spectra  of  terrestrial  substances,  enable  us  to  establish 
the  presence  of  certain  of  these  substances  in  the  stars, 
precisely  as  was  done  in  the  case  of  the  sun.  In  Aldebaran, 
for  instance,  the  presence  of  sodium,  magnesium,  tellurium, 
calcium,  antimony,  iron,  bismuth,  and  mercury,  has  been 
detected;  in  Sirius,  of  sodium,  magnesium,  iron,  and  hydrogen; 
in  Betelgeuse,  of  magnesium,  sodium,  calcium,  bismuth  and 
many  other  elements. 

The  diversity  of  brightness  in  the  stars  may  be  due  either 
to  a  difference  in  their  distances  from  us,  or  to  a  difference  in 
their  actual  dimensions.  Probably  both  causes  exist;  but 
it  is  fair  to  conclude  that,  as  a  general  rule,  the  brightest 
stars  are  the  nearest  to  us.  Observations  made  for  the 
purpose  of  determining  the  distances  of  the  stars  go  to 
justify  such  a  conclusion;  although  they  also  show  that 
the  rule  is  not  an  absolute  one,  since  some  of  the  fainter  stars 
are  found  to  be  nearer  to  us  than  some  of  the  brighter  ones. 
273.  Distance  of  the  Fixed  Stars. — No  perceptible  differ- 
ence is  detected  in  the  position  of  a  star  when  observed  at 


DISTANCE  OF  FIXED  STARS  247 

places  of  widely  different  latitudes.  The  conclusion  drawn 
from  this  fact  is,  that  the  stars  are  so  distant  that  lines  drawn 
from  any  two  points  on  the  earth's  surface  to  the  some  star 
are  sensibly  parallel:  in  other  words,  that  the  stars  have  no 
geocentric  parallax.  To  determine  the  distance  of  the  stars, 
then,  we  must  have  recourse  to  their  heliocentric  parallax. 
In  Fig.  75  let  S  be  the  sun,  AE'BE  the  orbit  of  the  earth,  s 
the  position  of  a  fixed  star,  supposed  to  lie  in  the  plane  of  the 
ecliptic,  and  NM  a  portion  of  the  celestial  sphere.  From 
s  draw  lines  sE,  sE',  tangent  to  the  earth's  orbit,  and  also 
prolong  them  beyond  s,  until  they  meet  the  arc  of  the  celestial 
sphere  NM.  Draw  the  radii  vectores  SE'  and  SE  to  the 


FIG.  75. 

points  of  tangency.  If  we  suppose  the  star  to  be  at  rest,  it  will 
lie  in  the  direction  E's,  when  the  earth  is  at  Ef,  and  in  the 
direction  Es,  when  the  earth  is  at  E,  and  the  motion  of  the 
earth  about  the  sun  will  give  the  star  an  apparent  oscillatory 
movement  over  the  arc  eef.  The  true  heliocentric  direc- 
tion in  which  the  star  lies  is  Ss' ;  and  the  difference  of  the 
directions  in  which  the  star  lies  at  any  time  from  the  sun  and 
the  earth  is  its  heliocentric  parallax.  This  difference  of  di- 
rection is  evidently  at  its  maximum  when  the  earth  is  at  Ef  or 
E;  and  this  maximum,  or  the  angle  SsE,  is  called  the  annual 
heliocentric  parallax,  or  simply  the  annual  parallax. 

Numerous  attempts  have  been  made  to  determine  the 
annual  parallax  of  the  stars,  by  comparing  observations 
made  when  the  earth  is  at  E  and  Ef.  The  nicest  observation, 


248  THE  FIXED  STARS.    NEBULA 

however,  has  failed  to  detect  in  any  star  a  parallax  as  great  as 
1";  though  recent  photographic  observations  have  definitely 
determined,  within  narrow  limits  the  parallaxes  of  a  large 
number  of  stars. 

274.  The  distance  of  the  fixed  stars,  then,  is  so  great  that 
the  radius  of  the  earth's  orbit,  92,900,000  miles,  does  not 
subtend  an  angle  of  even  I"  at  that  distance.     If,  in  the 
triangle  SsE,  we  suppose  the  angle  SsE  to  be  equal  to  1",  we 
shall  have, 

Ss  =  92,900,000  cosec  1": 

which  will  be  found  to  be  about  nineteen  trillions  of 
miles.  This  is  only  the  inferior  limit  of  the  distance  of 
the  stars:  that  is  to  say,  whatever  the  distance  may  be, 
it  cannot  be  less  than  this;  but  how  much  greater  it  may 
be,  particularly  in  the  case  of  those  numerous  stars  for 
which  no  parallaxes  can  be  detected,  it  is  impossible  to 
calculate. 

275.  Immensity  of  this  Distance. — It  is  hardly  possible 
to  obtain  a  clear  conception  of  a  distance  of  nineteen  trillions 
of  miles.     Perhaps  the  nearest  approach  to  such  a  conception 
is  made  by  considering  that  light,  moving  with  a  velocity  of 
186,000  miles  a  second,  and  passing  from  the  sun  to  the 
earth  in  a  little  more  than  eight  minutes,  would  consume 
about  3 \  years  in  accomplishing  such  a  distance :  so  that  when 
we  look  at  the  brightest  stars  in  the  heavens,  we  see  them,  not 
as  they  are  now,  but  as  they  were  3^  years  ago;  and  if  any 
one  of  them  were  to  be  destroyed  at  any  instant,  we  should 
continue  to  see  its  image  for  three  years  and  more  after  that 
time. 

Before  such  distances,  the  dimensions  of  the  solar  system 
shrink  to  the  insignificance  of  a  mere  point  in  space.  Nep- 
tune, the  most  distant  of  the  planets,  is  nearly  three  billions 
of  miles  from  the  sun;  and  yet,  if  Neptune  and  the  sun 
could  both  be  seen  from  the  nearest  fixed  star,  the  angular 
distance  between  them  would  never  be  greater  than  about 


DIFFERENTIAL  OBSERVATIONS  249 


30",  which  is  only  about  ^th  of  the  angle  which  the  sun 
subtends  to  us. 

276.  Differential  Observations.  —  In  dealing  with  so  small 
a  quantity  as  the  annual  parallax  of  the  stars,  it  is  important 
to  avoid  all  circumstances  by  which  even  the  most  minute 
errors  may  be  entailed  upon  the  observations.  The  apparent 
position  of  a  star  is  affected,  not  only  by  parallax,  but  by 
precession,  nutation,  aberration,  and  the  star's  own  proper 
motion.  The  laws  of  precession  and  nutation  enable  us  to 
decide  what  amount  of  the  apparent  motion  is  due  to  them; 
and  the  effect  of  a  star's  proper  motion  is  also  readily  sepa- 
rated from  that  of  parallax,  since  the  former  changes  the 
position  of  the  star  from  year  to  year,  while  the  latter  only 
changes  its  position  during  the  year,  causing  it  to  lie  now  on 
one  side  and  now  on  the  other  of  its  true  position,  but  giving 
it  no  annual  progressive  motion.  But  it  is  not  so  easy  to 
separate  the  effects  of  aberration  and  parallax.  Aberration, 
as  we  have  already  seen  (Art.  125),  causes  a  star  to  describe 
a  circle,  an  ellipse,  or  an  arc,  about  its  true  position  as  a 
center,  according  to  its  situation  with  reference  to  the  plane 
of  the  ecliptic  ;  and  it  is  easy  to  see  that  the  parallactic  move- 
ment of  a  star  is  of  precisely  the  same  character.  Thus  we 
have  already  seen,  in  Fig.  75,  that  a  star  situated  in  the  plane 
of  the  ecliptic  will  oscillate  by  parallax  over  the  arc  ee-  and 
if  the  star  is  not  in  the  plane  of  the  ecliptic,  lines  drawn  from 
all  points  of  the  earth's  orbit  to  the  star,  and  thence  pro- 
longed to  the  celestial  sphere,  will  evidently  meet  the  sphere 
in  a  circle  if  the  star  is  at  the  pole  of  the  ecliptic,  and  in  an 
ellipse  if  it  is  not  at  the  pole. 

In  order  to  separate  the  effects  of  parallax  and  aberration. 
the  following  method  was  adopted  by  the  astronomer  Bessel. 
Instead  of  attempting  to  determine  by  direct  observation  the 
change  of  position  of  the  star  whose  parallax  was  sought,  he 
selected  another  star  of  much  less  magnitude,  and  therefore 
supposed  to  be  at  a  much  greater  distance,  which  lay  very 
nearly  in  the  same  direction  as  the  first  star,  and  observed 


250 


THE  FIXED  STARS.    NEBULA 


the  changes  in  the  distance  between  these  two  stars  and  in 
the  direction  of  the  line  joining  them,  during  the  year.  Fig. 
76  will  serve  to  explain  the  general  principle  of  this  method. 
Let  S  be  the  position  of  the  star  whose  parallax  is  sought,  and 
s  the  position  of  the  smaller  star,  both  being  projected  on  the 
surface  of  the  celestial  sphere.  By  the  motion  of  the  earth 
in  its  orbit  the  star  S  will  describe  the  parallactic  ellipse 

A  DEC,  and  the  star  s,  the  ellipse 
adbc.  When  S  appears  to  be  at 
A,  s  will  appear  to  be  at  a;  when 
S  is  at  D,  s  will  be  at  d,  etc.  It  is 
evident  that  Aa  and  Bb  will  lie  in 
different  directions,  and  that  Dd 
will  be  greater  than  Cc:  and, 
therefore,  by  observing  the  dif- 
ferent directions  of  the  line  joining 
the  two  stars,  and  also  its  different 
values,  during  the  year,  we  may 
obtain  the  difference  of  parallax  of 
the  two  stars,  and,  approximately, 
the  parallax  of  S. 

277.  Results. — This  method  was  applied  by  Bessel  to  the 
star  61  Cygni.  For  the  sake  of  greater  accuracy  he  made  use 
of  two  very  small  stars,  situated  very  near  to  that  star,  whose 
absolute  parallaxes  he  assumed  to  be  equal.  The  parallax 
which  he  obtained  for  this  star  was  0".35.  Later  observa- 
tions give  a  value  of  0".30;  and  we  see  by  Art.  274  that  this 
corresponds  to  a  distance  which  light  would  require  about 
11.8  years  to  traverse.  >In  measuring  stellar  distance  the 
unit  employed  is  one  light-year  which  is  the  distance  that 
light  travels  in  one  year.  The  parallaxes  of  a  large  number 
of  other  stars  have  been  obtained  with  considerable  accuracy 
in  recent  years  by  photographic  and  spectroscopic  methods. 
The  star  which  is  nearest  to  the  earth  is  a  Centauri  and  latest 
observations  assign  it  a  parallax  of  0".750  which  corresponds 
to  a  distance  of  4.35  light-years. 


FIG.  76. 


REAL  DIMENSIONS  OF  THE  STARS  251 

According  to  the  Russian  astronomer  Peters,  the  mean 
parallax  of  the  stars  of  the  first  magnitude  is  0".21,  corre- 
sponding to  a  distance  which  light  would  traverse  in  15| 
years.  Another  Russian  astronomer,  Struve,  concludes 
that  the  distance  of  the  most  remote  stars  which  can  be  seen 
in  Lord  Rosse's  great  telescope  is  about  420  times  the  distance 
of  the  stars  of  the  first  magnitude :  from  which  the  marvelous 
inference  is  drawn  that  the  distance  of  the  most  remote 
telescopic  stars  from  the  earth  is  only  traversed  by  light  in 
6500  years. 

A  knowledge  of  the  distance  of  a  star,  and  of  its  proper 
motion,  enables  us  to  estimate  the  amount  in  miles  of  its 
transverse  motion.  Sirius,  for  instance,  has  a  proper  motion 
of  1".25  a  year,  and  its  parallax  is  0".38.  Hence  its  annual 
transverse  motion  is  equal  in  amount  to  the  radius  of  the 
earth's  orbit  multiplied  by  ^ff-:  which  is  a  motion  of  about 
ten  miles  a  second.  This  is  only  the  projection  upon  the  celes- 
tial sphere  of  its  real  motion,  which  may  be  much  greater. 

278.  Real  Dimensions  of  the  Stars. — Hitherto,  when  we 
have  determined  the  distance  of  a  celestial  body,  we  have 
been  able  to  compute  its  real  diameter  by  means  of  observa- 
tions made  upon  its  angular  diameter.  But  this  method  fails 
when  we  attempt  to  make  use  of  it  in  obtaining  the  dimen- 
sions of  the  stars,  since  they  do  not  present  any  measurable 
disc.  It  is  true  that  with  the  better  class  of  telescopes  some 
of  the  stars  appear  to  have  a  sensible  disc;  but  this  disc  is 
really  what  is  called  a  spurious  one.  This  is  proved  by  the 
fact  that  when  a  star  is  occulted  by  the  moon,  the  size  and 
the  shape  of  the  apparent  disc  remain  unaltered  up  to  the 
time  of  occultation,  and  its  disappearance  is  then  instanta- 
neous. In  the  case  of  a  solar  eclipse,  however,  or  of  the  occul- 
tation of  a  planet,  the  disappearance  of  the  disc  is  gradual. 

We  can,  however,  obtain  some  idea  of  the  probable 
diameter  of  a  star  by  comparing  the  light  which  it  emits 
with  that  which  is  emitted  by  the  sun.  This  comparison 
is  made  by  means  of  the  light  of  the  moon.  The  ratio  of  the 


252  THE  FIXED  STARS.     NEBULA 

light  of  the  sun  to  that  of  the  full  moon,  and  the  ratio  of  the 
light  of  the  full  moon  to  that  of  some  of  the  stars,  have  been 
obtained  by  appropriate  experiments;  and,  from  a  compari- 
son of  the  results  of  these  experiments,  it  is  inferred  that  if 
the  sun  were  removed  to  a  distance  from  the  earth  equal  to 
that  of  the  nearest  fixed  star,  it  would  appear  only  as  a  star 
of  the  second  magnitude.  The  probability,  then,  is  that 
unless  there  is  a  marked  difference  in  the  intensity  of  the 
light  which  these  different  bodies  emit  the  sun  is  not  so  large 
as  most,  and  perhaps  all,  of  the  stars  of  the  first  magnitude. 
There  is,  however,  as  might  be  expected  from  the  delicacy  of 
the  observations,  some  discrepancy  in  the  results  of  these 
various  photometric  experiments:  the  light  of  Sirius,  for 
instance,  is  said  by  some  observers  to  be  one  hundred,  and 
by  others  to  be  four  hundred,  times  as  great  as  the  light  of 
our  sun  would  be,  were  it  removed  to  a  distance  from  us 
equal  to  that  of  Sirius, 

VARIABLE  AND  TEMPORARY  STARS 

279.  Variable  Stars. — There  are  certain  stars  which 
exhibit  periodic  changes  in  their  brightness,  the  periods 
being  in  some  cases  only  a  few  days  in  length,  and  in  other 
cases  embracing  many  years.  The  star  o  Ceti,  called  also 
Mir  a,  is  an  example  of  this  class  of  stars.  When  brightest, 
it  is  a  star  of -the  second  magnitude.  It  remains  in  this 
state  for  about  two  weeks,  and  then  begins  to  diminish  in 
brightness,  becoming  wholly  invisible  to  the  naked  eye  in 
about  three  months,  and  appearing  in  telescopes  as  a  star  of 
the  ninth  or  the  tenth  magnitude.  After  about  five  months 
it  again  appears,  and  in  three  months  again  reaches  -its 
maximum  of  brightness.  The  period  in  which  these  changes 
occur  is  33 1J  days:  at  least,  that  is  its  mean  value:  its 
extreme  values  being  25  days  more  and  25  days  less 
than  this.  It  is  also  noticed  that  the  rate  of  its  increase  and 
decrease  of  brightness  is  not  always  the  same,  and  that  one 
maximum  of  brightness  is  not  always  equal  to  another. 


VARIABLE  AND  TEMPORARY  STARS  253 

Algol,  or  0  Persei,  is  another  remarkable  variable  star. 
It  remains  for  about  sixty-one  hours  as  a  star  of  the  second 
magnitude.  At  the  end  of  that  time  it  begins  to  decrease  in 
brightness,  and  becomes  a  star  of  the  fourth  magnitude  in 
less  than  four  hours.  After  about  twenty  minutes  its 
brightness  begins  to  increase,  and  another  period  of  less 
than  four  hours  brings  it  up  again  to  a  star  of  the  second 
magnitude. 

The  whole  number  of  variable  stars  is  more  than  3000. 

280.  Several  theories  have  been  advanced  in  explanation 
of  this  periodicity  of  brightness  in  the  variable  stars.     One 
theory  is  that  the  surfaces  of  these  stars  are  not  uniformly 
luminous,  and  that  therefore,  in  rotating  upon  their  axes, 
they  may  present  at  one  time  the  lighter  portions  of  their 
surfaces  to  the  earth,  and  the  darker  portions  at  another. 
Such  a  variation  of  luminosity  might  be  caused  by  the  pres- 
ence of  spots  on  the  surfaces  of  the  stars,  similar  to  the 
spots  on  the  sun,  but  of  much  greater  extent.      A  second 
theory,  known  as  the  eclipse  theory,  is  that  variable  stars 
are  really  double  stars  consisting  of  one  bright  and  one  or 
more  dark  stars;  when  the  dark  star  passes  between  the 
earth  and  the  bright  star  there  is  a  temporary  diminution  in 
the  light  that  we  receive.     Recent  observation  of  Algol  or 
j3  Persei  and  other  variables   practically  prove   that   the 
eclipse  theory  is  correct  in  the  case  of  a  certain  class  of 
variable  stars.     As  some  variables  have  an  irregular  period 
it  cannot  be  accepted  as  proved  that  all  of  these  variable 
stars   are   binaries.     Analogous   fluctuations   occur   in   the 
magnitude  and  period  of  the  sun's  spots  which  seem  to 
confirm    the    first    theory   for    variables    having    irregular 
periods.     Arago  suggests  that,  if  it  is  true,  as  has  been 
asserted  by  some  astronomers,  that  these  stars  when  at 
their  minimum  are  surrounded  by  a  kind  of  fog,  the  dimu- 
nution  of  light  may  be  due  to  the  interference  of  clouds. 

281.  Temporary    Stars. — There   are   stars   which   have 
appeared  at  times  in  different  parts  of  the  heavens,  and  have 


254  THE  FIXED  STARS.     NEBULAE 

afterwards  disappeared.  Such  stars  are  called  temporary 
stars.  In  comparing  recent  catalogues  of  stars  with  the 
catalogues  of  ancient  astronomers,  it  is  found  that  some  stars 
which  were  formerly  visible  are  no  longer  to  be  seen,  while 
others  which  are  now  visible  to  the  naked  eye  are  not  men- 
tioned in  the  ancient  catalogues.  Some  of  these  cases  may 
be  due  to  errors  of  observation,  but  hardly  all  of  them. 
Moreover,  similar  instances  have  occurred  in  modern  times. 
For  example,  it  is  recorded  by  the  Danish  astronomer,  Tycho 
Brahe,  that  in  November,  1572,  a  brilliant  star  suddenly 
blazed  forth  near  the  constellation  Cassiopeia,  and  remained 
in  sight  about  17  months.  When  at  its  brightest  phase,  it 
equaled  Venus  in  splendor,  and  was  visible  in  broad  daylight. 
It  disappeared  in  1574,  and  has  never  since  been  seen. 
Similar  temporary  stars  are  recorded  as  having  appeared 
in  or  near  the  same  place,  in  945  and  1264.  On  the  evening 
of  June  8,  1918,  a  star  in  the  constellation  Aquila  which  had 
been  of  the  llth  magnitude  was  noted  by  many  observers 
as  having  suddenly  increased  in  brightness.  Professor 
Barnard  states  that  this  star,  to  which  the  name  Nova 
Aquilse  III  of  1918  was  given,  was  brighter  than  Sirius. 
Photometric  determinations  of  the  magnitude  assigned  —0.5 
on  June  9  when  it  reached  its  maximum  and  from  then  on 
it  decreased  in  brightness  until  on  December  10,  its  mag- 
nitude had  decreased  to  5.67.  "  In  its  decline  it  showed  a 
sharply  defined  planetary  disk  wholly  different  from  the 
image  of  any  ordinary  star."  (Barnard.)  The  spectrum 
of  Nova  Aquilse  by  Professor  Parkhurst  showed  narrow  dark 
lines  in  a  continuous  band;  as  the  star  decreased  in  bright- 
ness the  spectrum  changed,  assuming  the  same  character  as 
spectra  of  other  bright  novae  with  a  few  very  broad  bright 
lines  accompanied  by  narrow  absorption  lines. 

Sir  John  Herschel  says,  with  reference  to  these  stars, 
that  it  is  worthy  of  notice  that  all  of  them  which  are  on 
record  have  been  situated  in  or  near  the  borders  of  the 
Milky  Way. 


DOUBLE  AND  BINARY  STARS  255 


DOUBLE  AND  BINARY  STARS 

282.  Double  Stars. — Many  of  the  stars  which  appear 
single  to  the  naked  eye  are  found,  when  examined  in  tele- 
scopes, to  consist  of  two  stars,  apparently  very  near  to  each 
other.  These  are  called  double  stars.  Only  four  were 
known  to  exist  until  near  the  close  of  the  last  century,  when 
Sir  William  Herschel  discovered  about  500.  The  whole 
number  of  double  stars  now  known  exceeds  13,000.  In  some 
cases  the  two  stars  are  nearly  of  the  same  magnitude,  but 
more  frequently  one  of  them  is  a  large  star,  and  the  other  a 
small  one.  Castor  is  an  instance  of  the  former  class,  and 
Sirius,  Vega,  and  Polaris  are  instances  of  the  latter  class. 
Some  stars  are  found  to  consist  of  three,  four,  five  or  more 


Castor.         Rigel.      "Solaris.      I  Cassiop.  12  Lyncis.      e  Lyrae. 


stars,  and  are  called  triple,  quadruple,  etc.,  stars.  The 
star  e  Lyrae,  for  instance,  appears  in  ordinary  telescopes  to 
consist  of  two  stars,  but  with  telescopes  of  greater  power 
each  of  these  stars  is  resolved  into  two  others. 

283.  Binary  Stars. — The  question  arises  with  reference 
to  these  combinations:  are  the  stars  which  compose  them 
really  connected,  as  are  the  sun  and  the  planets;  or  is  their 
appearance  merely  an  optical  illusion,  arising  from  the  fact 
that  the  stars  in  any  one  combination  happen  to  lie  in  the 
same  direction  from  the  earth,  although  they  may  at  the 
same  time  be  at  an  immense  distance  from  each  other?  The 
chances,  at  all  events,  are  very  much  against  the  latter 
supposition.  The  astronomer  Struve  has  calculated  that 
there  is  only  about  one  chance  in  9570  that,  if  the  stars  of 
the  first  seven  magnitudes  were  scattered  at  random  in  the 
heavens,  any  two  of  them  would  fall  within  4"  of  each  other; 
and  yet  more  than  100  such  cases  have  been  observed.  He 


256  THE  FIXED  STARS.     NEBULA 

has  further  calculated  that  there  is  only  about  one  chance  in 
200,000  that  three  stars  would  accidentally  fall  within  30" 
of  each  other,  so  as  to  form  a  triple  star;  and  at  least  four 
such  cases  are  to  be  found. 

The  chances,  then,  are  that  most  of  the  stars  in  these  vari- 
ous combinations  are  physically  connected.  But  more  than 
this:  it  was  announced  by  Sir  William  Herschel  in  1803,  after 
twenty-five  years  of  observation  upon  many  of  the  double 
stars,  that  in  each  double  star  which  he  had  examined,  the 
two  stars  of  which  it  was  composed  revolved  about  each  other 
in  regular  orbits,  and  in  fact  constituted  a  sidereal  system. 
Subsequent  observations  by  other  astronomers  have  fully 
verified  this  conclusion,  and  about  600  double  stars  have 
been  found  to  consist  of  stars  revolving  about  each  other, 
or  rather  about  their  common  center  of  gravity,  according 
to  the  Newtonian  law  of  gravitation.  Such  double  stars 
are  called  binary  stars,  to  distinguish  them  from  other  double 
stars,  the  components  of  which  have  not  as  yet  been  found 
to  be  physically  connected.  There  are  also  other  double 
stars,  the  components  of  which,  while  as  yet  they  do  not 
seem  to  revolve  about  each  other,  have  constantly  the  same 
proper  motion:  thus  showing  that  they  are  in  all  probability 
moving  as  one  system  through  space.  Triple,  quadruple, 
etc.,  stars,  whose  constituents  are  found  to  be  physically  con- 
nected, are  called  ternary,  quarternary,  etc.,  stars. 

The  orbits  of  the  binary  stars  usually  are  found  to  be 
ellipses  of  considerable  eccentricity.  The  periods  of  their 
revolutions  have  also  been  approximately  determined,  and 
extend  over  a  very  wide  range.  There  are  only  about  twelve 
stars  whose  periods  are  less  than  100  years,  and  only  about 
150  whose  periods  are  less  than  1000  years. 

284.  Alpha  Centauri. — The  star  a  Centauri,  a  star  of  the 
first  magnitude  in  the  southern  hemisphere,  is  found  to  con- 
sist of  two  components,  one  of  the  first  magnitude  and  the 
other  of  the  second.  The  relative  positions  of  these  two 
components  have  been  carefully  noted  during  recent  years. 


COLORED  STARS 


257 


1826^ 


In  Fig.  77,  A  represents  one  of  those  components,  and 
B  Bf  B"  the  apparent  path  of  the  other  about  it.  The 
major  axis  of  the -orbit  is  about  40",  and  the  period  81.2 
years.  Its  eccentricity  is  0.63. 

Since  the  distance  of  a  Centauri  from  the  earth  is  approx- 
imately known,  we  can  obtain  some  idea 
of  the  dimensions  of  the  orbit.  If  E  in 
the  figure  represents  the  earth,  we  shall 
have,  in  the  right-angled  triangle  COE, 
the  angle  E  equal  to  20",  and  the  side 
EO  equal  to  21  trillions  of  miles.  Hence, 

CO  =  0#Xtan  20"  =  2,000,000,000  miles: 

which  is  about  equal  to  the  distance  of 
Uranus  from  the  sun,  or  to  22  times  the 
radius  of  the  earth's  orbit. 

Again,  knowing  the  radius  of  the  orbit 
and  the  period,  we  can  obtain  an  approxi- 
mate value  of  the  mass  of  a  Centauri 

from  the  formula  in  Art.  214.     It  will  be  found  to  be  about 

twice  the  mass  of  the  sun. 

COLORED  STARS 

286.  Many  of  the  stars,  both  isolated  and  double,  shine 
with  a  colored  light.  The  isolated  colored  stars  are  usually 
red  or  orange:  blue  or  green  stars  being  very  uncommon. 
The  number  of  red  stars  now  recognized  is  at  least  300,  about 
40  of  which  are  visible  to  the  naked  eye.  Among  the  most 
conspicuous  of  the  red  stars  are  Aldebaran  and  Antares ;  and 
it  is  worthy  of  notice  that  nearly  all  the-  variable  stars  of 
long  period  are  of  this  color. 

The  components  of  the  double  stars  are  often  of  different 
colors;  blue  and  yellow,  or  green  and  yellow;  and,  less  fre- 
quently, white  and  purple,  or  white  and  red.  The  following 
table  contains  a  few  of  the  many  instances  of  such  stars  which 
might  be  given: 


258 


THE  FIXED  STARS.     NEBULA 


Name. 

Magnitude 
of 
Components. 

Color  of  the  Larger. 

Color  of  the  Smaller. 

17  Cassiopeise  
a  Piscium  
i  Cancri  
e  Bootis 

4.  ...7 
5.  ...6 
5.  ...8 
3        7 

Yellow 
Pale  green 
Orange 
Pale  orange 

Purple 
Blue 
Blue 
Sea  green 

f  Coronse  
0  Cygni  

5   ...6 
3.  ...7 

White 
Yellow 

Purple 
Blue 

When  the  colors  of  the  components  are  complementary, 
and  the  components  are  of  very  unequal  size,  it  is  possible 
that  only  one  of  the  colors  may  really  exist ;  the  other  being, 
according  to  a  law  of  Optics,  merely  the  result  of  contrast. 
Such  an  opinion  is  held  by  some  astronomers;  but  the  objec- 
tion is  raised  to  it  by  others  that,  if  one  of  these  colors  is  only 
accidental,  it  ought  to  disappear  when  the  eye  is  shielded  from 
the  light  of  the  star  which  has  the  other  color:  which,  how- 
ever, is  very  far  from  being  the  case.  Another  objection  is 
that  a  similar  phenomenon  ought  to  be  seen  in  all  colored 
stars  whose  components  are  of  unequal  sizes:  whereas  many 
double  stars  are  found  in  which  both  components  have  the 
same  color,  sometimes  red  and  sometimes  blue.  In  one  of 
Struve's  catalogues,  out  of  596  double  stars,  there  were: 

295  pairs,  both  white; 
118  pairs,  both  yellowish  or  reddish; 
63  pairs,  both  bluish; 
120  pairs  of  totally  different  colors. 

In  a  few  instances  stars  appear  to  have  changed  their 
color.  Sirius,  for  instance,  is  now  white;  but  both  Ptolemy 
and  Seneca  expressly  state  that  in  their  day  it  had  a  reddish 
hue.  Capella,  which  is  now  yellow,  was  formerly  red.  Some 
observers  state  that  seventeen  stars  of  the  first  magnitude  are 
colored,  and  that  seven  of  these  have  changed  their  color. 
This  change  of  color  is  particularly  noticeable  in  the  tempo- 


CLUSTERS  AND    NEBULA  259 

rary  stars.  In  that  of  1572  (Art.  281),  the  color  changed  from 
white  to  yellow,  and  then  to  a  decided  red ;  and  it  is  observed 
generally  in  the  variable  stars  that  the  redness  of  the  light 
increases  as  its  intensity  diminishes. 

Spectroscopic  observations  seem  to  show  that  the  color 
of  the  stars  is  dependent  to  a  large  extent  on  their  stage  of 
cosmic  development.  Sirius,  Vega,  and  the  other  white  and 
blue  stars  show  strong  evidence  of  the  presence  of  hydrogen 
and  are  considered  to  have  evolved  from  nebulse  within  a 
comparatively  recent  period.  Capella  and  Pollox  show  lines 
in  their  spectra  much  like  those  of  the  sun;  stars  of  this 
class  are  yellow  and  are  believed  to  be  of  almost  the  same 
age  as  our  sun.  The  third  class  includes  most  of  the  red 
and  variable  stars  and  their  spectra  are  characterized  by 
dark  broadened  bands  instead  of  lines,  though  there  are 
generally  lines  also;  Aldebaran,  Mira,  and  Betelgeuse  (a 
Orionis)  are  examples  of  this  class.  The  faint  red  stars  form 
a  fourth  class  with  spectra  sometimes  containing  a  few  bright 
lines;  these  are  apparently  the  oldest  stars.  This  determina- 
tion of  the  age  of  stars  is  purely  hypothetical  and  the  present 
stage  of  the  science  does  not  enable  astronomers  to  be  sure 
which  are  youngest  or  oldest. 

CLUSTERS  AND  NEBULAE 

286.  If  we  examine  the  heavens  on  any  clear  night  when 
the  moon  is  below  the  horizon,  we  shall  find  here  and  there 
groups  of  stars,  which  present  a  hazy,  cloud-like  appearance. 
These  groups  are  classified  into  clusters  and  nebulce,  although 
it  is  difficult  to  establish  any  precise  distinction  between  the 
two  classes.  When  the  different  members  of  a  group,  or  at 
all  events  some  of  them,  can  be  separated  from  each  other 
with  the  telescope,  the  group  is  called  a  cluster.  A  nebula, 
on  the  other  hand,  is  either  wholly  invisible  to  the  naked  eye, 
or,  if  seen  at  all,  presents,  as  the  name  implies,  only  an  ill- 
defined,  cloudy  appearance.  The  number  of  nebulae  which 


260  THE  FIXED  STARS.     NEBULA 

have  been  discovered  is  many  thousands.  Some  of  the  so- 
called  nebulas  are  resolved,  with  the  aid  of  the  telescope, 
into  separate  stars :  while  others,  even  when  examined  in  the 
most  powerful  telescopes,  preserve  their  cloudy  appearance, 
and  give  no  trace  of  stars;  the  latter  are  true  nebulae. 

As  every  increase  of  telescopic  power  has  rendered  resolv- 
able some  of  the  nebulas  previously  classified  as  irresolvable, 
the  common  opinion  of  astronomers,  until  within  the  last 
few  years,  was  that  there  was  no  distinction  between  these 
nebulae  more  fundamental  than  that  of  distance,  dimension, 
or  intensity  of  light.  Recently,  however,  the  light  of  some 
of  these  nebulas  has  been  subjected  to  spectroscopic  analysis, 
and  the  results  seem  to  show  the  existence  of  decidedly 
different  classes  of  nebulas.  The  researches  of  Mr.  Huggins 
show  that  the  light  of  some  of  these  nebulae  gives  spectra, 
which  are  distinguished,  not  by  dark  lines,  but  by  bright 
ones:  which  shows,  as  we  have  already  noticed  (Art.  102), 
that  the  light  does  not  come  from,  solid  bodies  in  a  state  of 
ignition,  but  from  incandescent  vapor  or  gas.  One  class  of 
nebulae  is  undoubtedly  gaseous;  evidence  of  the  spectroscope 
on  this  point  is  conclusive ;  the  presence  of  hydrogen  is  well 
established  and  the  other  constituents  seem  to  vary  with 
different  nebulae.  The  Great  Nebula  in  Orion  is  an  instance 
of  the  gaseous  class  of  nebulae. 

The  nebulae  are  very  unequally  distributed  over  the 
heavens.  They  congregate  especially  in  a  zone  crossing 
the  Milky  Way  at  right  angles,  and  they  are  especially 
abundant  in  the  constellations  Leo,  Ursa  Major,  and  Virgo. 

287.  Examples  of  Clusters. — The  most  noticeable  cluster 
is  the  well-known  group  called  the  Pleiades.  This  group 
contains  six  or  seven  stars  which  are  visible  to  the  naked 
eye,  the  brightest  star  being  of  the  third  magnitude:  and 
glimpses  of  many  more  can  be  obtained  by  examining  the 
group  with  the  eye  turned  sideways.  The  telescope  and 
photography  reveal  hundreds  and  even  thousands  of  stars 
belonging  to  the  group. 


CLUSTERS  AND  NEBULA  261 

The  Hyades  are  a  group  in  Taurus,  near  the  star  Alde- 
baran,  in  which  from  five  to  seven  stars  can  be  counted.  This 
group  was  supposed  by  the  ancients  to  have  some  influence 
upon  the  rain. 

Praesepe,  or  the  Bee-hive,  in  Cancer,  is  visible  to  the  naked 
eye  as  a  luminous  spot.  With  a  telescope  of  moderate  power, 
more  than  forty  conspicuous  stars  are  seen  within  a  space 
about  30'  square,  together  with  many  smaller  ones.  Before 
the  invention  of  the  telescope,  this  group  was  considered  a 
nebula. 

288.  Different  Forms  of  Nebulae. — The  groups  which 
usually  go  by  the  name  of  nebulae  present,  as  might  be 
expected,  almost  every  variety  of  form;  and  most  of  these 
forms  are  too  irregular  to  admit  of  any  classification  or 
description.     Such  is  not  the  case,  however,  with  all  of  them: 
and  some  of  the  different  varieties  of  shape  are  exemplified  in 
the  following  list: 

1.  Annular  Nebulae; 

2.  Elliptic  Nebulae; 

3.  Spiral  Nebulae; 

4.  Planetary  Nebulae; 

5.  Nebulous  stars. 

There  are  also  individual  nebulae  whose  names  indicate 
the  general  appearance  which  they  present:  such  are  the 
"Horse-shoe  Nebula,"  the  "Crab  Nebula,"  the  "Dumb-bell 
Nebula,"  etc. 

289.  Annular  Nebulae. — Of   annular      or      ring-shaped 
nebulae,  the  heavens  present  several  examples.     The  most 
remarkable  one  is  situated  in  the  northern  constellation  Lyra. 
Sir  John  Herschel  says  of  it:  "It  is  small,  and  particularly 
well  defined,  so  as  to  have  more  the  appearance  of  a  flat, 
oval,  solid  ring  than  of  a  nebula.     The  axes  of  the  ellipse  are 
to  each  other  in  the  proportion  of  about  four  to  five,  and  the 
opening  occupies  rather  more  than  half  the  diameter.     The 
central  vacuity  is  filled  in  with  faint  nebulae,  like  a  gauze 


262  THE  FIXED  STARS.    NEBULA 

stretched  over  a  hoop.  The  powerful  telescopes  of  Lord 
Rosse  resolve  this  object  into  excessively  minute  stars,  and 
show  filaments  of  stars  adhering  to  its  edges." 

290.  Elliptic  Nebulae. — There  are  several   instances   of 
elliptic  or  oval  nebulae,  of  various  degrees  of  eccentricity. 
The  very  conspicuous  nebula  called  "The  Great  Nebula  in 
Andromeda"  is  an  example  of  this  class.     :c  This  object  is 
distinctly  visible  to  the  naked  eye,  and  is  sometimes  mistaken 
for  a  comet.     When  viewed  with  a  telescope  of  moderate 
power,  it  has  an  elongated  oval  form ;  but  in  the  largest  tele- 
scopes its  aspect  is  very  different  from  this.     Professor  G.  P. 
Bond  traced  it  through  a  length  of  4°,  and  a  breadth  of  2J0, 
and  also  discovered  in  it  two  curious  black  streaks,  lying 
nearly  parallel  to  the  major  axis  of  the  oval.     He  also  suc- 
ceeded in  detecting  in  it  evidence  of  a  stellar  constitution. 

Latest  observations  of  the  Andromeda  nebula  show  clear 
evidence  of  spiral  structure. 

Some  elliptic  nebulae  are  remarkable  for  having  double 
stars  at  or  near  each  of  their  foci. 

291.  Spiral   Nebulae. — Observations    with    the    Earl    of 
Rosse's  telescope  have  led  to  the  discovery  of  nebulae  which 
consist  of  spiral  bands  proceeding  from  a  common  nucleus, 
and  sometimes  from  two  nuclei.     The  most  remarkable  of 
these  spiral  nebulae  is  situated  near  the  extremity  of  the  tail 
of  the  Great  Bear.     It  consists  of  nebulous  coils  diverging 
from  two  luminous  centers,  about  5'  apart,  and  gives  evidence 
of  stellar  composition. 

292.  Planetary    Nebulae. — Planetary    nebulae    received 
their  name  from  Sir  William  Herschel,  on  account  of  the 
resemblance  in  form  which  they  bear  to  the  larger  planets  of 
our  system.     The  outlines  of  some  of  them  are  well  defined, 
while  those  of  others  are  indistinct;  and  some  of  them  shine 
with  a  blue  light.     About  twenty-five  have  been  discovered, 
most  of  them  being  situated  in  the  southern  hemisphere. 
One  of  these  nebulae  is  situated  near  /3  Ursae  Majoris,  and  has 
a- diameter  of  2'  40".     It  was  discovered  by  Mechain,  in  1781, 


CLUSTERS  AND  NEBULA  263 

and  is  described  as  "a  very  singular  object,  circular  and  uni- 
form, which  after  a  long  inspection  looks  like  a  condensed 
mass  of  attenuated  light."  Perforations  and  a  spiral  ten- 
dency have  been  detected  in  it  by  the  Earl  of  Rosse. 

These  planetary  nebulae  are  not  globular  clusters  of 
stars,  as  they  would  in  that  case  be  brighter  in  the 
middle  than  at  the  borders,  instead  of  being,  as  they  are, 
uniformly  bright  throughout.  The  spectroscope  definitely 
determines  that  the  planetary  nebula?  have  a  gaseous  con- 
stitution. 

293.  Nebulous  Stars. — Nebulous  stars  are  stars  which 
are  surrounded  by  a  faint  nebulous  envelope,  usually  circular 
in  form,  and  sometimes  several  minutes  in  diameter.     In 
some  cases  this  envelope  terminates  in  a  distinct  outline, 
while  in  others  it  gradually  fades  away  into  darkness.     Ac- 
cording to  Hind,  nebulous    stars  "have    nothing   in  their 
appearance  to  distinguish  them  from  others  entirely  desti- 
tute  of  such  appendages;   nor  does  the  nebulous  matter 
in  which   they  are  situated  offer  the  slightest  indications 
of  resolvability  into  stars,  with  any  telescopes  hitherto  con- 
structed." 

294.  Double    Nebulae.— M.    D'Arrest,   of  Copenhagen, 
mentions  fifty  double  nebulse  whose  components  are  not  more 
than  5'  apart,  and  estimates  that  there  may  be  as  many  as 
200  such  double  nebula?.     The  two  components  are  generally 
of  a  globular  form. 

It  is  possible  that  these  components  may  in  some  cases 
be  physically  connected.  In  one  instance  there  seem  to  have 
been  changes  in  the  distance  and  the  relative  position  of  the 
two  members  during  the  last  eighty  years,  which  indicate 
the  possibility  of  a  revolution  of  one  of  the  members  about  the 
other. 

295.  Magellanic  Clouds. — These  two  nebulse,  called  also 
nubecula  major  and  nubecula  minor,  are  situated  near  the 
southern  pole  of  the  heavens.     They  are  visible  to  the  naked 
eye  at  night,  when  the  moon  is  not  shining,  and  have  an  oval 


264  THE  FIXED  STARS.     NEBULAE 

shape.  One  of  them  covers  an  area  of  forty-two  square 
degrees,  the  other  an  area  of  ten  square  degrees.  Sir  John 
Herschel  found  them  to  consist  of  swarms  of  stars,  clusters, 
and  nebula?  of  every  description. 

296.  Variation  of  Brightness  in  the  Nebulae. — In  the  case 
of  a  few  of  the  nebulae  a  change  of  brightness  has  been  dis- 
covered, which  is  to  some  extent  analogous  to  what  has 
already  been  noticed  in  connection  with  the  variable  and  the 
temporary  stars.  In  1852  Mr.  Hind  discovered  a  small 
nebula  in  the  constellation  Taurus.  This  nebula  was 
observed  from  that  time  until  the  year  1856;  but  in  1861  it 
had  entirely  disappeared.  In  1862  it  was  seen  in  the  tele- 
scope at  Pulkowa,  and  for  a  short  time  appeared  to  be 
increasing  in  brightness;  but  towards  the  end  of  1863,  Mr. 
Hind,  upon  searching  for  it  with  the  telescope  with  which 
it  was  originally  discovered,  was  unable  to  find  any  trace  of 
it.  It  is  a  curious  coincidence  that  a  star  situated  very  near 
to  this  nebula  has  changed,  in  the  same  interval  of  time, 
from  the  tenth  magnitude  to  the  twelfth. 

There  are  four  or  five  instances  on  record  of  similar 
changes  in  the  brightness  of  nebula.  The  nebula  known  as 
80  of  Messier's  Catalogue,  is  said  to  have  changed,  in  the 
months  of  May  and  June,  1860,  from  a  nebula  to  a  well- 
defined  star  of  the  seventh  magnitude,  and  then  to  have 
resumed  its  original  appearance.  The  change  was,  however, 
so  rapid  and  so  decided,  that  some  astronomers  are  inclined 
to  ascribe  it  to  the  existence  of  a  variable  star  between  the 
nebula  and  the  earth,  rather  than  to  a  variation  in  the 
nebula  itself.  The  Great  Nebula  in  the  constellation  Argo, 
when  observed  by  Sir  John  Herschel  in  1838,  contained  in  its 
densest  part  the  star  77,  which  was  then  of  the  first  magnitude. 
Observations  made  in  1863,  however,  showed  that  this  star 
had  become  entirely  free  from  its  nebulous  envelope,  and 
had  also  been  reduced  to  a  star  of  the  sixth  magnitude.  The 
outline  of  certain  parts  of  the  nebula  was  also  found  to  be 
different  from  what  it  had  been  represented  to  be  by  Herschel. 


CLUSTERS  AND  NEBULA  265 

More  recent  observations  show  still  further  changes  in  this 
nebula. 

297.  The  Galaxy,  or  Milky  Way.— The  Milky  Way,  that 
well-known  luminous  band  which  stretches  through  the 
heavens,  may  be  considered  to  be  a  continuous  resolvable 
nebula.  If  we  follow  the  line  of  its  greatest  brightness,  and 
neglect  occasional  deviations,  we  find  its  course  to  be  nearly 
that  of  a  great  circle  of  the  heavens,  inclined  to  the  equinoctial 
at  an  angle  of  about  63°.  At  one  point  a  kind  of  branch  is 
sent  off,  which  unites  again  with  the  main  stream  at  a  dis- 
tance of  about  150°  from  the  point  of  separation.  The 
angular  breadth  of  the  belt  varies  from  6°  to  20°,  the  average 
breadth  being  about  10°.  When  examined  with  the  tele- 
scope, this  band  is  found  to  be  made  of  thousands  and  mil- 
lions of  telescopic  stars,  grouped  together  with  every  degree 
of  irregularity.  Of  the  20,000,000  stars  of  the  first  fourteen 
magnitudes,  about  nine-tenths  are  in  or  near  the  Milky  Way. 
At  some  points  the  stars  are  so  close  to  each  other  that  they 
form  one  bright  mass  of  light ;  while  at  other  points  there  are 
dark  spaces  containing  scarcely  a  star  which  is  visible  to  the 
naked  eye.  A  very  noticeable  break  of  this  kind  occurs  in 
that  part  of  the  Milky  Way  which  lies  nearest  to  the  south 
pole.  This  dark  space  is  about  8°  in  length  and  5°  in  breadth. 
It  contains  only  one  star  visible  to  the  naked  eye,  and  that 
is  a  very  small  one.  Early  southern  navigators  gave  to  this 
vacancy  the  name  of  the  Coal  Sack. 

298.  Number  of  the  Stars. — Scarcety  more  can  be  said 
of  the  probable  number  of  the  stars,  than  that  it  is  as  incon- 
ceivably great  as  are  the  distances  by  which  they  are  sep- 
arated from  the  earth  and  from  each  other.  Sir  William 
Herschel  states  that  on  one  occasion,  while  observing  the 
stars  in  the  Milky  Way,  he  estimated  that  116,000  stars 
passed  through  the  field  of  his  telescope  in  fifteen  minutes; 
and  that,  on  another  occasion,  nearly  260,000  stars  passed  in 
forty-one  minutes.  It  may  assist  us  in  correctly  appreciating 
the  immensity  of  these  numbers  to  remember  that  the 


266  THE  FIXED  STARS.     NEBULA 

number  of  stars  visible  to  the  naked  eye  in  the  northern  hem- 
isphere is  about  7000  (Art.  266). 

MOTION  OF  THE  SOLAR  SYSTEM  IN  SPACE 

299.  We  have  now  accustomed  ourselves  to  consider  the 
earth  to  be  a  planet,  rotating  upon  a  fixed  axis  once  in  every 
sidereal  day,  and  revolving  about  the  sun  once  in  every 
sidereal  year;  but  we  have  up  to  this  point  considered  the 
sun  and  the  solar  system  to  be  at  rest  in  space.  There  are, 
however,  observations  which  show  that  such  is  not,  in 
truth,  the  case,  but  that  the  whole  solar  system  is  in  rapid 
motion  through  space.  Analogy  certainly  supports  such  a 
conclusion.  We  have  already  seen  that  the  solar  system, 
immense  as  it  may  seem  to  be  in  itself,  must  be  considered  as 
scarcely  more  than  a  point  in  comparison  with  the  entire 
celestial  system  (Art.  275);  and  we  have  also  seen  reasons 
for  concluding  (Art.  278)  that  there  are  many  bodies  among 
the  stars  which  are  very  much  larger  than  the  sun.  There 
is,  then,  no  reason  for  supposing  that  the  solar  system  is  the 
center  of  the  celestial  system;  and  there  is  nothing  violent  in 
the  conclusion  that,  just  as  Jupiter  revolves  about  the  sun, 
and  carries  its  satellites  with  it,  so  the  sun  may  revolve 
about  some  other  body  or  point,  and  carry  its  system  of 
planets  and  satellites  with  it. 

300.  The  Apparent  Motions  of  the  Stars. — The  proper 
motions  of  the  stars,  already  mentioned,  may  be  explained 
in  three  distinct  ways.  First,  they  may  be  what  they  seem 
to  be,  real  motions  of  the  stars,  performed  in  orbits  of  im- 
mense radii;  or,  secondly,  they  may  be  only  apparent  motions, 
caused  by  a  change  of  the  position  of  the  solar  system  in 
space;  or,  thirdly,  they  may  be  the  results  of  both  these 
causes  existing  together.  The  last  of  these  theories  was 
advanced  by  Sir  William  Herschel,  in  1783;  and  although 
doubts  were  afterwards  expressed  as  to  its  truth,  later 
observations  have  tended  to  support  it,  and  it  is  now  gener- 


MOTION  OF  THE  SOLAR  SYSTEM  IN  SPACE        267 

ally  accepted  by  astronomers.  If  we -suppose  for  a  moment 
that  the  solar  system  is  approaching  any  given  point  in  the 
heavens,  the  pole-star,  for  instance,  the  result  will  be  that 
the  stars  which  lie  about  that  point  will  appear  to  separate 
slowly  from  it,  while  the  stars  which  lie  abouyt  the  diametric- 
ally opposite  point  will  appear  to  close  up  around  that 
point;  and  indeed  all  the  stars  will  apparently  move  in  an 
opposite  direction  to  that  in  which  we  suppose  the  solar 
system  to  be  moving,  those  stars  having  the  greatest  motion 


which  lie  in  a  direction  at  right  angles  to  the  direction  of  the 
motion  of  the  solar  system.  In  Fig.  78,  let  E  be  the  position 
in  space  of  the  earth  at  any  time,  and  let  P,  A,  B,  D,  and  G 
be  stars  supposed  to  be  situated  at  equal  distances  from  the 
earth.  Let  the  earth,  by  the  motion  of  the  whole  solar 
system  in  space,  be  carried  to  some  point  E',  in  the  direction 
of  the  star  P.  It  is  evident  that  the  angular  distance 
of  the  star  A  from  P,  when  the  earth  is  at  E',  or  the  angle 
PE'A,  is  greater  than  the  angular  distance  between  A  and 


268  THE  FIXED  STARS.     NEBULAE 

P  when  the  earth  is  at  E,  or  the  angle  PEA.  In  other  words, 
while  the  earth  has  moved  from  E  to  E',  the  star  A  has 
apparently  receded  from  P.  So  also  the  star  D  has  appar- 
ently approached  the  star  C.  The  stars  B  and  G  have  both 
receded  from  P,  the  retrogradation  of  G,  or  the  angle  EGE', 
being  greater  than  the  retrogradation  of  B,  or  the  angle 
EBE'. 

301.  Direction  and  Amount  of  this  Motion. — The  ele- 
ments, then,  of  the  motion  of  the  solar  system  through  space 
are  determined  from  what  are  called  the  proper  motions  of 
the  stars.  Recent  observations  and  calculations  have  not 
only  confirmed  Herschel's  theory  that  such  a  motion  really 
exists,  but  have  also  very  nearly  confirmed  his  conclusion 
as  to  the  point  towards  which  the  motion  is  directed.  Differ- 
ent astronomers  give  different  positions  to  this  point;  but 
they  all  agree  in  the  general  conclusion,  that  the  solar  system 
is  at  present  moving  in  the  direction  of  a  point  in  the  con- 
stellation Hercules,  situated  in  about  17J  hours  of  right 
ascension,  and  35°  of  north  declination. 

The  calculations  of  M.  Otto  Struve  led  him  to  the 
conclusion  that  for  a  star  situated  at  right  angles  to  the 
direction  of  the  motion  of  the  solar  system  and  at  the  mean 
distance  of  the  stars  of  the  first  magnitude,  the  annual 
angular  displacement  of  the  star  due  to  that  motion  is 
0//34:  that  is  to  say,  the  distance  through  which  the  system 
moves  in  one  year  subtends  at  the  star  an  angle  of  that 
amount.  Now,  the  mean  parallax  of  the  stars  of  the  first 
magnitude,  or  the  angle  subtended  at  the  mean  distance  of 
those  stars  by  the  radius  of  the  earth's  orbit,  is  estimated  to 
be  O."21  (Art.  277);  hence  the  annual  motion  of  the  solar 
system  is  the  radius  of  the  earth's  orbit  multiplied  by  f  f ,  or 
about  150,000,000  miles:  a  little  less  than  five  miles  a  second. 

Mr.  Airy,  however,  deduced  a  motion  of  about  twenty- 
seven  miles  a  second,  while  Mr.  Campbell's  researches  show 
a  velocity  of  about  12  miles  per  second,  and  this  is  the  value 
commonly  accepted  by  astronomers. 


MOTION  OF  THE  SOLAR  SYSTEM  IN  SPACE        269 

302.  The   Orbit  of  the   Solar   System. — Although   the 
motion  of  the  solar  system  through  space  appears  to  be 
rectilinear,  it  does  not  follow  that  such  is  actually  the  case : 
since  it  may  move  in  an  elliptical  orbit,  with  a  radius  vector 
so  immense  that  years  and  even  centuries  of  observation  will 
be  needed  to  show  any  sensible  change  of  direction.     The 
probability  is  that  the  sun,  with  its  planets  and  their  satel- 
lites, revolves  about  the  common  center  of  gravity  of  the 
group  of  stars  of  which  it  is  a  member;  and  that  this  center 
of  gravity  is  situated  in  or  near  the  plane  of  the  Milky  Way. 
If  we  suppose  the  orbit  in  which  our  system  is  moving  to  be 
an  ellipse  with  a  small  eccentricity,  then  the  center  of  this 
ellipse  will  lie  in  a  direction  which  makes  an  angle  of  about 
90°  with  the  direction  in  which  the  system  is  now  moving. 
Madler  concluded  that  Alcyone,  the  brightest   star  in  the 
Pleiades,  was  the  central  sun  of  the  celestial  sphere,  while 
Argelander,  believing  that  this  central  point  must  lie  in  the 
principal  plane  of  the  Milky  Way,  places  it  in  the  constella- 
tion Perseus.    It  may  be  noticed,  in  connection  with  this  sub- 
ject, that  Sir  William  Herschel  was  inclined  to  believe  that 
some  of  the  more  conspicuous  of  the  stars,  such  as  Sirius, 
Arcturus,  Capella,  Vega,  and  others,  were  in  a  great  degree 
out  of  the  reach  of  the  attractive  power  of  the  other  stars, 
and  were  probably,  like  the  sun,  centers  of  systems. 

303.  Years,  and,  in  all  probability,  ages  of  observation 
will  be  needed  to  determine  the  elements  of  this  most  stu- 
pendous orbit.     Madler's  speculations  have  led  him  to  the 
conclusion  that  the  period  of  this  revolution  is  29,000,000 
years:  a  period  in  comparison  with  which  the  years  through 
which  astronomical  observations  have  extended  are  utterly 
insignificant.     The  student  who  is  curious  to  know  more  of 
this  subject  will  find  the  details  fully  set  forth  in  Grant's 
History   of  Physical   Astronomy.     Grant   himself  says,   in 
reference  to  the  subject:    "It  is  manifest  that  all  such 
speculations  are  far  in  advance  of  practical  astronomy,  and 
therefore  they  must  be  regarded  as  premature,   however 


270  THE  FIXED  STARS.     NEBULA 

probable  the  supposition  on  which  they  are  based,  or  how- 
ever skillfully  they  may  be  connected  with  the  actual  obser- 
vation of  astronomers." 

DETERMINATION  OF  THE  REAL  MOTIONS  OF  THE  STARS 

304.  We  have  already  seen  (Art.  265),  that  in  order  to 
determine  the  real  motion  of  a  star  in  space,  we  must  be  able 
to  determine,  not  only  its  transverse  motion,  which  is  indi- 
cated by  a  change  in  its  apparent  position  upon  the  celestial 
sphere,  but  also  its  motion  directly  to  or  directly  from  the 
earth.     Now,  a  star  situated  at  the  nearest  distance  of  the 
fixed  stars,  and  moving  towards  the  earth  with  a  velocity 
equal  to  that  of  the  earth  in  its  orbit  (18  miles  a  second), 
would  diminish  its  distance  from  us  by  only  about  -faih  in  a 
thousand  years.     The  detection  of  any  such  motion  by  a 
change  in  the  star's  apparent  brightness,  is  therefore,  utterly 
out  of  the  question.     The  spectroscope,  however,  gives  us 
quite  another  means  of  detecting  such  a  motion. 

305.  Analogy    between    Light    and    Sound. — A    clear 
conception  of  the  principle  upon  which  this  method  rests  may 
be  more  readily  obtained  if  we  first  notice  the  analogy  be- 
tween the  conduction  of  light  and  that  of  sound.     Sound  is 
the  result  of  a  series  of  waves  or  pulses  in  the  air,  produced 
by  the  vibrations  of  a  sonorous  body;  light  is  the  result  of  a 
similar  series  of  pulses  or  waves  in  the  luminiferous  ether, 
caused  by  the  vibrations  of  the  particles  of  a  luminous  body. 
The  more  rapidly  a  sonorous  body  vibrates,  the  greater  will 
be  the  number  of  pulses  or  waves  which  it  communicates  to 
the  air  in  the  unit  of  time,  and,  consequently,  the  higher  will 
be  the  pitch  of  the  tone  produced.     In  the  case  of  a  luminous 
body,  the  greater  the  number  of  waves  in  the  luminiferous 
ether  which  the  vibrations  of  any  particle  cause  in  the  unit 
of  time,  the  greater  will  be  the  refrangibility  of  the  ray 
produced;  in  the  solar  spectrum,  for  instance,  the  violet 
rays^are  the  most  refrangible  of  all  the  rays  which  we  can 


REAL  MOTIONS  OF  STARS  271 

see,  and  the  number  of  waves  in  the  unit  of  time  necessary 
to  produce  them  is  greater  than  the  number  necessary  to 
produce  rays  of  any  other  color.  The  refrangibility  of  a  ray 
is  therefore  analogous  to  the  pitch  of  a  tone. 

306.  Change  of  Tone  or  Refrangibility. — It  is  proved  by 
experiment  that  if  the  distance  between  a  sonorous  body, 
producing  a  tone  of  constant  pitch,  and  the  hearer,  is  dim- 
inished by  the  motion  of  either,  with  a  velocity  that  has  an 
appreciable  ratio  to  that  of  sound,  the  pitch  of  the  tone  will 
appear  to  be  heightened;  and  that  if,  on  the  contrary,  the 
distance  between  the  two  is  increased  with  sufficient  rapidity, 
the  pitch  of  the  tone  will  appear  to  be  lowered.     So,  too,  in 
the  case  of  light:  if  the  luminous  body  and  the  observer 
approach  each  other,  the  refrangibility  of  the  rays  of  light 
will  be  increased;  and  if  they  separate,  it  will  be  diminished. 
If,  then,  we  can  detect  any  change  of  refrangibility  in  the 
light  of  a  heavenly  body,  we  may  conclude  that  the  distance 
of  that  body  from  the  earth  is  changing. 

307.  Change  of  Refrangibility  Detected. — Father  Secchi, 
in  a  communication  to  the  Comptes  Rendus  in  the  early  part 
of  1868,  after  stating  the  principles  of  this  method,  announced 
that  he  had  subjected  the  light  of  Sirius  and  of  other  prom- 
inent stars  to  an  examination  with  the  spectroscope,  but  had 
detected  no  evidence  of  motion.     After  that,  Mr.  Huggins, 
an  English  observer,  who  has  devoted  himself  especially  to 
spectroscopic    investigations,    was    successful    in    detecting 
such  a  motion  in  Sirius.     There  is  a  certain  dark  line  F  in 
the  blue  of  the  solar  spectrum,  which  corresponds  to  a  bright 
line  in  the  spectrum  of  hydrogen;  and  this  same  line  also 
appears  in  the  spectrum  of  Sirius.     Now,  if  Sirius  has  no 
motion  either  towards  or  from  the  earth,  the  line  F  in  its 
spectrum  will  coincide  in  position  with  the  corresponding  line 
in  the  spectrum  of  hydrogen,  when  the  two  spectra  are 
compared  by  means  of  the  spectroscope:  while  if,  on  the 
contrary,   Sirius  has  such  a  motion,   its  whole  spectrum, 
lines  and  all,  will  be  shifted  bodily,  and  the  line  F  will  no 


272  THE  FIXED  STARS.     NEBULAE 

longer  coincide  with  the  corresponding  line  in  the  spectrum 
of  hydrogen. 

Mr.  Huggins,  using  a  spectroscope  of  large  dispersive 
power,  and  carefully  comparing  the  spectrum  of  Sirius  with 
that  of  hydrogen,  found  that  the  line  F  in  the  spectrum  of 
Sirius  was  displaced,  by  about  ^i^th  of  an  inch.  This  dis- 
placement was  towards  the  red  end  of  the  spectrum,  and 
indicated  that  the  refrangibility  of  the  light  of  Sirius  was 
diminished:  since  the  red  rays  are  the  least  refrangible  of  all 
the  colored  rays  of  the  spectrum.  Sirius  was  therefore 
receding  from  the  earth. 

308.  Amount  of  the  Real  Motion  of  Sirius. — The  amount 
of  recession  corresponding  to  a  displacement  of  this  extent, 
when  observed  in  a  spectroscope  whose  power  is  equal  to 
that  of  the  one  used  by  Mr.  Huggins,  is  computed  to  Be  about 
41J  miles  a  second.  But  it  happens  that  when  the  observa- 
tion was  made,  the  earth  was  so  situated  in  its  orbit  that  it 
was  receding  from  Sirius,  by  its  revolution  in  its  orbit,  at  the 
rate  of  about  12  miles  a  second.  If  the  motion  of  the 
solar  system  in  space,  is  taken  to  be  12  miles  a  second 
(Art.  301),  this  must  also  be  taken  into  consideration; 
and  the  point  towards  which  this  motion  was  directed  was 
almost  exactly  opposite  to  the  position  of  Sirius  on  the  celes- 
tial sphere.  The  earth  was  therefore  moving  away  from  Sirius 
at  the  rate  of  about  24  miles  a  second.  If,  then,  we  diminish 
the  whole  amount  of  the  increase  of  distance  between  Sirius 
and  the  earth  through  space  in  one  second,  by  the  amount  of 
the  motion  of  the  earth  through  space  in  one  second  or  24 
miles,  we  find  that  Sirius  was  moving  through  space,  away 
from  the  earth,  at  the  rate  of  about  17 J  miles  a  second. 

By  combining  this  motion  with  the  transverse  motion 
of  Sirius,  we  can  obtain  an  approximate  value  of  its  real 
motion.  In  Art.  277,  the  transverse  motion  of  Sirius  was 
computed  to  be  about  16  miles  a  second.  The  resultant  of 
these  two  motions  is  about  29  miles  a  second :  or,  900,000,000 
miles  a  year. 


REAL  MOTIONS  OF  THE  STARS  273 

Later  observations  show  that,  between  1876  and.  1884, 
Sirius  ceased  to  recede  from  the  earth,  and  is  now  approach- 
ing it  at  about  4  miles  per  second;  and  the  other  stars  are 
found  to  be  moving  from  or  towards  the  earth. 

309.  The  numerical  results  of  the  preceding  article  may 
not  be  beyond  criticism;  but  the  grand  fact  remains,  that  in 
all  probability  the  motions  of  these  distant  bodies,  which 
have  so  long  seemed  to  be  wrapped  in  hopeless  mystery,  are 
soon  to  come  within  the  reach  of  our  observation.  The 
knowledge  of  these  motions  will  be  a  powerful  auxiliary  in 
the  determination  of  the  law  which  undoubtedly  binds  all 
the  heavenly  bodies  together  in  one  great  system;  and  it  is 
not  presumptuous  to  expect  that  at  some  future  day — no  one 
can  say  how  distant  or  how  near — this  law  will  be  revealed 
to  us. 


APPENDIX 


MATHEMATICAL    DEFINITIONS    AND    FORMULA 

PLANE  TRIGONOMETRY 

1.  The  complement  of  an  angle  or  arc  is  the  remainder  ob- 
tained by  subtracting  the  angle  or  arc  from  90°. 

2.  The  supplement  of  an  angle  or  arc  is  the  remainder 
obtained  by  subtracting  the  angle  or  arc  from  180°. 

3.  The  reciprocal  of  a  quantity  is  the  quotient  arising 
from  dividing  1  by  that  quantity:  thus  the  reciprocal  of  a 

is?  _  c>| 

4.  In  the  series  of  right  triangles  ABC,    ( 
AB'C',   AB"C",  etc.,  Fig.   79,   having   a 
common  angle  A,  we  have  by  Geometry, 

BC    B'C'    B"C" 


O/^Y          E>'/"y          D"/^" 
X5  vy  .£5    vy  X5      Vy 

~AC  =  ^Cf==~ACrr] 

AB  =  ABf  =AB" 
AC~AC'~~  AC"' 


FIG.  79. 


The  ratios  of  the  sides  to  each  other  are  therefore  the 
same  in  all  right  triangles  having  the  same  acute  angle:  so 
that,  if  these  ratios  are  known  in  any  one  of  these  triangles, 
they  will  be  known  in  all  of  them.  These  ratios,  being  thus 
dependent  only  on  the  value  of  the  angle,  without  any  regard 

275 


276  APPENDIX 

to  the  absolute  lengths  of  the  sides,  have  received  special 
names,  as  follows: 

The  sine  of  the  angle  is  the  quotient  of  the  opposite  side 

-nri 

divided  by  the  hypothenuse.     Thus,  sin  A  = 


The  tangent  of  the  angle  is  the  quotient  of  the  opposite 

TIC* 

side  divided  by  the  adjacent  side.     Thus,  tan  A=^-^. 

AC 

The  secant  of  the  angle  is  the  quotient  of  the  hypothenuse 
divided  by  the  adjacent  side.     Thus,  sec  A=-—^. 


5.  The  cosine,  cotangent,  and  cosecant  of  the  angle  are 
respectively  the  sine,  tangent,  and  secant  of  the  complement 
of  the  angle.  Now,  in  Fig.  79,  the  angle  ABC  is  evidently 
the  complement  of  the  angle  BAG.  Hence  we  have, 

AC 
cos  A  =  sin  B  =  -r-=; 

AC 


AB 

cosec  A  =  sec  B  =       . 


6.  Since  the  reciprocal  of  -  is  -,  we  see,  from  the  preceding 

definitions,  that  the  sine  and  the  cosecant  of  the  same  angle, 
the  tangent  and  the  cotangent,  the  cosine  and  the  secant,  are 
reciprocals. 

7.  If  two  parts  of  a  plane  right  triangle  in  addition  to  the 
right  angle  are  given,  one  of  them  being  a  side,  the  triangle 
can  be  solved:  that  is  to  say,  the  values  of  the  remaining 
parts  can  be  obtained.  This  solution  is  effected  by  means 
of  the  definitions  above  given. 

8.  When  an  angle  is  very  small,  its  sine  and  its  tangent 
are  both  very  nearly  equal  to  the  arc  which  subtends  the 
angle  in  the  circle  whose  radius  is  unity.  Hence,  to  find  the 


APPENDIX  277 

sine  or  the  tangent  of  a  very  small  angle  approximately,  we 
have,  if  #  is  a  small  angle  expressed  in  seconds, 

sin  x  =  tan  x  =  x  sin  I". 
If  x  is  expressed  in  minutes, 

sin  x  =  tan  x  =  x  sin  1'. 

9.  If  x  and  y  are  any  two  small  angles,  we  have  from  the 
preceding  article, 

sin  x _x  sin  V _x 
sin  y     y  sin  \"     y' 

that  is,  the  sines  (or  tangents)  of  very  small  angles  are  propor- 
tional to  the  angles  themselves. 

10.  Cos  x=l-2  sin2  \  x. 

11.  The  sides  of  a  plane  triangle  are  proportional  to  the 
sines  of  their  opposite  angles. 

12.  In  order  to  solve  a  plane  oblique  triangle,  three  parts 
must  be  given,  and  one  of  them  must  be  a  side. 

SPHERICAL  TRIGONOMETRY 

13.  A  spherical  triangle  is  a  triangle  on  the  surface  of  a 
sphere,  formed  by  the    arcs  of  three    great  circles  of  the 
sphere. 

14.  In  a  spherical  right  triangle,  the  sine  of  either  oblique 
angle  is  equal  to  the  quotient  of  the  sine  of 

the  opposite  side,  divided  by  the  sine  of 
the  hypothenuse.  Thus,  in  the  triangle 
ABC,  right-angled  at  C,  Fig.  80,  we  have, 

sin  BC 

sin  A  =  - — -7-=.  FIG.  80. 

sin  AB 

15.  The  cosine  of  either  oblique  angle  is  equal  to  the  quo- 
tient of  the  tangent  of  the  adjacent  side,  divided  by  the 
tangent  of  the  hypothenuse.     Thus, 

tan  AC 

cos  A  =  r     -7-=. 
tan  AB 


278 


APPENDIX 


16.  The  tangent  of  either  oblique  angle  is  equal  to  the 
quotient  of  the  tangent  of  the  opposite  side,  divided  by  the 
sine  of  the  adjacent  side.  Thus, 


tan  A  — 


tan  BC 
sin  AC' 


17.  A  spherical  right  triangle  can  be  solved  when  any  two 
parts  in  addition  to  the  right  angle  are  given.     The  solution 
is  effected  by  means  of  the  formulae  given  in  the  preceding 
articles. 

ANALYTIC  GEOMETRY 

18.  The  circle,  the  ellipse,  the  hyperbola,  and  the  para- 
bola are  often  called  conic  sections;  since  each  curve  may  be 
formed  by  intersecting  a  right  cone  by  a  plane. 

19.  An  ellipse  is  a  plane  curve,  in  which  the  sum  of  the 

distances  of  each  point  from  two 
fixed  points  is  equal  to  a  given  line. 
Thus,  in  Fig.  81,  the  sum  of  the 
distances  of  the  point  M  from  the 
two  fixed  points  F  and  Fr  is  always 
constant,  wherever  on  the  curve 
ACBD  the  point  M  may  be  sit- 
uated. 

The  two  fixed  points  are  called  the  foci  of  the  ellipse,  and 
the  middle  point  of  the  line  which  joins  them  is  called  the 
center. 

A  line  drawn  from  either  focus  to  any  point  of  the  curve 
is  called  a  radius  vector. 

A  line  drawn  through  the  center,  and  terminating  at  each 
extremity  in  the  curve,  is  called  a  diameter.  That  diameter 
which  passes  through  the  foci  is  called  the  transverse  or 
major  axis:  and  that  diameter  drawn  perpendicular  to  the 
transverse  axis  is  called  the  conjugate  or  minor  axis.  Thus, 
AB  is  the  transverse  axis,  and  CD  the  conjugate  axis.  The 


APPENDIX 


279 


transverse  axis  is  equal  to  the  constant  sum  of  the  distances 
FM  and  F'M. 

The   eccentricity  of  the  ellipse  is  the   quotient  of  the 
distance  from  the  center  to  either  focus,  divided  by  half  the 

OF 
major  axis.     Thus,  -^-=  is  the  eccentricity. 

20.  The  solid  generated  by  the  revolution  of  an  ellipse 
about  its  major  axis  is  called  a  prolate  spheroid:  that  generated 
by  its  revolution  about  its  minor  axis  is  called  an  oblate 
spheroid. 

The  expression  for  the  volume  of  an  oblate  spheroid  is 
jpra2  b:  in  which  a  and  6  denote  the  semi-major  and  the  semi- 
minor  axis  of  the  gen- 
erating ellipse,  and  TT 
the  ratio  of  the  circum- 
ference of  a  circle  to  its 
diameter,  or  3.1416. 

The  compression,  or 
oblateness,  of  an  oblate 
spheroid  is  the  ratio  of 
the  difference  between 
the  major  and  the  minor 
axis  of  the  generating  ellipse  to  the  major  axis. 

21.  The  hyperbola  is  a  plane  curve,  in  which  the  difference 

of  the  distances  of  each  point  from  two 
fixed  points  is  equal  to  a  given  line.  Thus, 
in  Fig.  82,  the  difference  of  the  distances 
of  any  point  M  of  the  curve  from  the 
two  fixed  points  F  and  F'  is  equal  to  a 
given  line. 

The  two  fixed  points  are  called  the 
foci,  and  the  middle  point  of  the  line 
FIG  83  joining  them  is  called  the  center. 

22.  The   parabola  is   a  plane  curve, 

every  point  of  which  is  equally  distant  from  a  fixed  point,  and 
from  a  right  line  in  the  same  plane.     Thus,  in  Fig.  83,  in 


280  APPENDIX 

which  AB  is  the  given  line,  and  F  the  given  point,  the  dis- 
tances FM  and  GM  are  equal  to  each  other,  for  any  point  M 
of  the  curve. 

The  fixed  point  is  called  the  focus. 

MECHANICS 

23.  The  resultant  of  two  or  more  forces  is  a  force  which 
singly  will  produce  the  same  mechanical  effect  as  the  forces 
themselves  jointly. 

The  original  forces  are  called  components.  In  all  statical 
investigations  the  components  may  be  replaced  by  their 
resultant,  and  vice  versa. 

24.  If  two  forces  be  represented  in  magnitude  and  direc- 
tion by  the  two  adjacent  sides  of  a  parallelogram,  the  diagonal 
will  represent  their  resultant  in  magnitude  and  direction. 

Thus,  in  Fig.  84,  if  a  force 
act  at  A  in  the  direction 
AD',  and  a  second  force  act 
at  A  in  the  direction  A  B', 
these  two  forces  being  rep- 
resented in  magnitude  by 
the  lengths  of  the  lines  AD 
and   A  B  respectively,   the 
resultant  of  these  two  forces 
will  be  a  force  in  the  direc- 
tion AC',  and  of  a  magnitude  represented  by  the  length  of 
the  line  AC.     The  parallelogram  is  called  the  parallelogram 
of  forces. 

25.  Conversely :  any  given  force  may  be  resolved  into  two 
component  forces,  acting  in  given  directions.     Thus,  in  Fig. 
84,  let  AC  be  the  given  force,  and  AB'  and  AD'  the  given 
directions.     From  C  draw  CB  parallel  to  AD',  and  CD 
parallel  to  AB',  AB  and  AD  will  be  the  two  components 
acting  in  the  given  directions. 

26.  The  force  which  must  continually  urge  a  material 
point  towards  the  center  of  a  circle,  in  order  that  it  may 


APPENDIX  281 

describe  the  circumference  with  a  uniform  velocity,  is  equal 
to  the  square  of  the  linear  velocity  divided  by  the  radius. 
This  proposition  is  expressed  in  the  following  formula: 

mv2 


m  being  the  mass,  v  the  space  passed  over  in  the  unit  of 
time,  r  the  radius  of  the  circle,  and  /  the  magnitude  of  the 
force. 

27.  Curvilinear  motion  of  a  body  is  the  result  of  a  con- 
tinuous force,  the  direction  of  which  is  different  from  the 
direction  of  the  original  impulse.     This  force,   called   the 
deflecting  force,  can  be  resolved  into  two  components,  —  one 
normal  to  the  path  of  the  body,  and  the  other  tangential  to 
it.     The  normal  component  is  called  the  centripetal  force. 
The  moving  body,  through  inertia,  resists  this  force,  the 
resistance  being  equal  and  opposite  in  direction  to  the  cen- 
tripetal force.     This  force  of  resistance  is  called  the   cen- 
trifugal force. 

28.  Let  t  be  the  periodic  time,  or  the  time  of  one  revolu- 
tion.    We  shall  evidently  have, 


Substituting  this  value  of  v2  in  the  formula  in  Art.  26  above. 
we  shall  have, 


282  APPENDIX 


ASTRONOMICAL  CHRONOLOGY 

THE  science  of  Astronomy  seems  to  have  been  cultivated 
in  very  early  ages  by  almost  all  the  Eastern  nations,  partic- 
ularly by  the  Egyptians,  the  Persians,  the  Indians,  and  the 
Chinese.  Records  of  observations  made  by  the  Chaldseans 
as  far  back  as  2234  B.C.  are  said  to  have  been  found  in  Baby- 
lon. The  study  of  Astronomy  was  continued  by  the  Chal- 
daeans,and  in  later  times  by  the  Greeks  and  the  Romans,  until 
about  A.D.  200.  After  that  time  it  was  neglected  for  about 
six  centuries,  and  was  then  resumed  by  the  Eastern  Saracens 
after  Bagdad  was  built.  It  was  brought  into  Europe  in  the 
thirteenth  century  by  the  Moors  of  Barbary  and  Spain.  A 
full  account  of  the  rise  and  the  progress  of  the  study  of 
Astronomy,  is  given  in  Vince's  Astronomy,  Vol.  II. 

The  instrument  principally  used  by  the  ancient  astrono- 
mers seems  to  have  been  a  sort  of  quadrant,  mounted  in  a 
vertical  position.  Ptolemy  describes  one  which  he  used 
in  determining  the  obliquity  of  the  ecliptic.  The  Arabian 
astronomers  had  one  with  a  radius  of  21f  feet,  and  a  sextant 
with  a  radius  of  57 f  feet. 

The  following  dates,  with  scarcely  any  change  or  addition, 
have  been  taken  from  George  F.  Chambers 's  Descriptive 
Astronomy  (Oxford,  1867),  in  which  many  other  interesting 
astronomical  dates,  here  omitted,  may  be  found.  (See 
Note,  page  289.) 

B.C. 

720.  Occurrence  of  a  lunar  eclipse,  recorded  by  Ptolemy.  It 
seems  to  have  been  total  at  Babylon,  March  19, 
9J  h.  P.M. 

640-550.  Thales,  of  Miletus,  one  of  the  seven  wise  men  of 
Greece.  He  declared  that  the  earth  was  round,  cal- 
culated solar  eclipses,  discovered  the  solstices  and  the 
equinoxes,  and  recommended  the  division  of  the  year 
into  365  days. 


APPENDIX  283 


B.C. 

585.  Occurrence  of  a  solar  eclipse,  said  to  have  been  predicted 
by  Thales. 

580-497.  Pythagoras,  of  Crotona,  founder  of  the  Italian 
Sect.  He  taught-  that  the  planets  moved  about  the 
sun  in  elliptic  orbits,  and  that  the  earth  was  round; 
and  is  said  to  have  suspected  the  earth's  motion. 

547.  Anaximander  died.  He  asserted  that  the  earth  moved, 
and  that  the  moon  shone  by  light  reflected  from  the 
sun.  He  introduced  the  sun-dial  into  Greece. 

500.  Parmenides,  of  Elis,  taught  the  sphericity  of  the  earth. 

490.  Alcmseon,  of  Crotona.  asserted  that  the  planets  moved 
from  west  to  east. 

450.  Diogenes,  of  Apollonia,  stated  that  the  inclination  of  the 
earth's  orbit  caused  the  change  of  seasons.  Anaxagoras 
is  said  to  have  explained  eclipses  correctly. 

432.  Meton  introduced  the  luni-solar  period  of  19  years.  He 
observed  a  solstice  at  Athens  in  424. 

384-322.  Aristotle  wrote  on  many  physical  subjects,  includ- 
ing Astronomy. 

370.  Eudoxus  introduced  into  Greece  the  year  of  365J  days. 

330.  Callippus  introduced  the  luni-solar  period  of  76  years. 
Pytheas  measured  the  latitude  of  Marseilles,  and 
showed  the  connection  between  the  moon  and  the  tides. 
He  also  showed  the  connection  of  the  inequality  of  the 
days  and  nights  with  the  differences  of  climate. 

320-300.  Autolychus,  author  of  the  earliest  works  on  Astron- 
omy extant  in  Greek. 

306.  First  sun-dial  erected  in  Rome. 

287-212.  Archimedes  observed  solstices,  and  attempted  to 
measure  the  sun's  diameter.  Aristarchus  declared  that 
the  earth  revolved  about  the  sun,  and  rotated  on  its 
axis. 

276-196.  Eratosthenes  measured  the  obliquity  of  the  ecliptic 
and  found  it  to  be  20J  degrees.  He  also  measured  a 
degree  of  the  meridian  with  considerable  exactness. 


284  APPENDIX 

B.C. 

190-120.  Hipparchus,  called  the  Newton  of  Greece.  He 
discovered  precession:  used  right  ascensions  and 
decimations,  and  afterwards  latitudes  and  longitudes: 
calculated  eclipses :  discovered  parallax :  determined  the 
mean  motions  of  the  sun  and  the  moon;  and  formed  the 
first  regular  catalogue  of  the  stars. 

50.  Julius  Caesar,  with  the  Egyptian  astronomer  Sosigenes, 
undertook  to  reform  the  calendar. 

A.D. 

100-170.  Ptolemy,  of  Alexandria,  author  of  the  Ptolemaic 
System  of  the  Universe,  in  which  the  earth  is  the  center. 
He  wrote  descriptions  of  the  heavens  and  the  Milky 
Way,  and  formed  a  catalogue  giving  the  positions  of 
1022  fixed  stars.  He  appears  to  have  been  the  first 
to  notice  the  refraction  of  the  atmosphere. 

582.  The  School  of  Astronomy,  at  Alexandria,  founded  ten 
centuries  previously  by  Ptolemy  the  Second,  was 
destroyed  by  the  Saracens  under  Omar. 

762.  Rise  of  Astronomy  among  the  Eastern  Saracens. 

880.  Albatani,  the  most  distinguished  astronomer  between 
Hipparchus  and  Tycho  Brahe.  He  corrected  the  values 
of  precession  and  the  obliquity  of  the  ecliptic,  formed  a 
catalogue  of  the  stars,  and  first  used  sines,  chords, 
etc. 

1000.  Abul-Wefa  first  used  secants,  tangents,  and  cotan- 
gents. 

1080.  The  use  of  the  cosine  introduced  by  Geber,  and  also 
some  improvements  in  Spherical  Trigonometry. 

1252.  Alphonso  X.,  King  of  Castile,  under  whose  direction 
certain  celebrated  astronomical  tables,  called  the 
Alphonsine  Tables,  were  compiled. 

1484.  Waltherus  used  a  clock  with  toothed  wheels.  (The 
earliest  complete  clock  of  which  there  is  any  certain 
record  was  made  by  a  Saracen  in  the  thirteenth 
century.) 


APPENDIX  285 

A.D. 

1543.  Publication  of  Copernicus's  De  Revolutionibus  Orbium 
Celestium,  setting  forth  the  Copernican  System  of  the 
Universe. 

1581.  Galileo  determined  the  isochronism  of  the  pendulum. 

1582.  Tycho  Brahe  commenced  astronomical  observations 
near  Copenhagen. 

1594.  Gerard   Mercator,    author   of   Mercator's   Projection. 

(The  date  is  doubtful,  and  may  have  been  as  early  as 

1556.) 
1576.  Fabricius  discovered  the  variability  of  o  Ceti. 

1603.  Bayer's  Maps  of  the  Stars  published. 

1604.  Kepler  obtained  an  approximate  value  of  the  correction 
for  refraction. 

1608.  Jansen  and  Lippersheim,  of  Holland,  are  said  to  have 
invented  the  refracting  telescope,  using  a  convex  lens. 
It  is,  however,  a  disputed  point  as  to  who  really  invented 
the  telescope.     The  use  of  the  lens  seems  to  have  been 
known  about  fifty  years  before  this. 

1609.  Kepler  announced  his  first  two  laws. 

1610.  Galileo,  using  a  telescope  with  a  concave  object-lens, 
discovered  the  satellites  of  Jupiter,  the  librations  of 
the  moon,  the  phases  of  Venus,  and  some  of  the  nebulae. 

1611.  Spots  and  rotation  of  the  sun  discovered  by  Fabricius. 
1614.  Napier  invented  logarithms. 

1618.  Kepler  announced  his  third  law. 

1631.  The  first  recorded  transit  of  Mercury,  observed  by 

Gassendi.     The  vernier  invented. 
1633.  Galileo  forced  to  deny  the  Copernican  theory. 

1639.  First  recorded  transit  of  Venus,  observed  by  Horrox 
and  Crab  tree. 

1640.  Gascoigne  applied  the  micrometer  to  the  telescope. 
1646.  Fontana  observed  the  belts  of  Jupiter. 

1654.  The  discovery  of  Saturn's  rings  by  Huyghens. 
(Galileo  had  previously  noticed  something  peculiar 
in  the  planet's  appearance.) 


286  APPENDIX 

A.D. 

1658.  Huyghens    made    the    first    pendulum-clock.     (The 
discovery  is  also  ascribed  to   Galileo  the  younger.) 

1662.  Royal  Society  of  London  founded. 

1663.  Gregory  invented  the  Gregorian  reflecting  telescope. 

1664.  Hook  detected  Jupiter's  rotation. 

1666.  Foundation  of  the  Academy  of  Sciences  at  Paris. 
1669.  Newton  invented  the  Newtonian  reflecting  telescope. 

1673.  Flamsteed  explained  the  equation  of  time. 

1674.  Spring  pocket-watches  invented  by  Huyghens.     (Said 
also  to  have  been  invented,  somewhat  earlier,  by  Dr. 
Hooke.) 

1675.  Transmission  of  light  discovered  by  Romer.     Transit 
Instrument   used    to    determine    right   ascensions    by 
Romer.     Greenwich  Observatory  founded. 

1687.  Newton's  Principia  published. 

1690.  Ellipticity  of  the  earth  theoretically  determined  by 

Huyghens. 

1704.  Meridian  Circle  used  by  Romer. 
1711.  Foundation  of  the  Royal  Observatory  at  Berlin. 

1725.  Compensation    pendulum    announced    by    Harrison. 

1726.  Mercurial  pendulum  invented  by  Graham. 

1727.  Aberration  of  light  discovered  by  Bradley. 
1731.  Hadley's  Quadrant  invented. 

1744.  Euler's  Theoria  Motuum  published,  the  first  analytical 
work  on  the  planetary  motions. 

1745.  Nutation  of  the  earth's  axis  discovered  by  Bradley. 
1750.  Wright's  Theory  of  the  Universe  published. 

1765.  Harrison  rewarded  by  Parliament  for  the  invention 

of  the  Chronometer. 

1767.  British  Nautical  Almanac  commenced. 
1769.  Transit  of  Venus  successfully  observed. 
1774.  Experiments  by  Maskelyne  to  determine  the  earth's 

density. 
1781.  Uranus  discovered  by  Sir  William  Herschel.     Messier's 

catalogue  of  Nebulae  published. 


APPENDIX  287 

A.D. 

1783.  Herschel  suspected  the  motion  of  the  whole  solar 
system. 

1784.  Researches  of  Laplace  on  the  stability  of  the  solar 
system. 

1786.  Publication  of  Herschel's  catalogue  of  1000  nebulae. 

1787.  Herschel  began  to  observe  with  his  forty-foot  reflector. 
The  Trigonometrical  Survey  of  England  commenced. 

1788.  Publication    of    La  Grange's  Mecanique  Analytique. 

1789.  The  rotation  of  Saturn  determined  by  Herschel,  and  a 
second  catalogue  of  1000  nebulae  published. 

1792.  Commencement   of   the   Trigonometrical   Survey   of 

France. 

1796.  French  Institute  of  Science  founded. 
1799.  Laplace's  Mecanique  Celeste  commenced. 
1801-7.  The  minor  planets  Ceres,  Pallas,  Juno  and  Vesta 

discovered. 

1802.  Publication  of  Herschel's  third  catalogue  of  Nebulae. 

1803.  Publication  of  Herschel's  discovery  of  Binary  Stars. 

1804.  Proper  motions  of  300  stars  published  by  Piazzi. 

1805.  Commencement  of  attempts  to  determine  the  parallax 
of  the  stars. 

1812.  Troughton's  Mural  Circle  erected  at  Greenwich. 

1820.  Foundation   of  the   Royal   Astronomical   Society   of 
London. 

1821.  Observatory  at  Cape  of  Good  Hope  founded. 
1823.  Cambridge  (England)  Observatory  founded. 
1835.  Airy  determined  the  time  of  Jupiter's  rotation. 

1837.  Value  of  the  moon's  equatorial  parallax  determined  by 
Henderson.     The  East  Indian  arc  of  21°  21'  completed. 

1838.  Parallax  of  61  Cygni  determined  by  Bessel. 

1839.  Parallax   of  a   Centauri   determined  by  Henderson. 
Imperial  Observatory  at  Pulkowa   (Russia)   founded. 

1840.  Harvard  Observatory  founded. 

1842.  Washington  Observatory  founded.     Mean  density  of 
the  earth  determined  by  Baily. 


288  APPENDIX 

A.D. 

1843.  Periodicity  of  the  solar  spots  detected  by  Schwabe. 

1844.  Taylor's  catalogue  of  11,015  stars.  Transmission  of 
time   by  electric   signals   commenced   in   the   United 
States. 

1845.  Discovery  of  the  fifth  minor  planet  Astraea.     (More 
than  800  others  have  since  been  discovered.) 

1846.  Discovery  of  the  planet  Neptune. 

1847.  Lalande's  catalogue  of  47,390  stars  republished  by  the 
British  Association. 

1851.  Foucault's  pendulum  experiment  to  demonstrate  the 
earth's  rotation.  Completion  of  the  Russo-Scandi- 
navian  arc. 

1854.  Difference  of  longitude  of  Greenwich  and  Paris  deter- 
mined by  electric  signals. 

1855.  Commencement  of  the  American  Ephemeris. 

1858.  De  La  Rue  obtained  a  stereoscopic  view  of  the  moon. 
(The  first  photograph  of  the  moon  was  made  by  Dr. 
J.  W.  Draper,  of  New  York,  in  1840.) 

1859.  Publication   of   Section   I.   of  Argelander's  "Zones," 
containing  110,982  stars.     Kirchoff's  observations  of 
color  spectra.     Completion  of  the  Berlin  Star  Charts 
commenced  in  1830. 

1861.  Appearance,  in  June,  of  a  comet  with  a  tail  of  105° 
(the  longest  on  record).     Publication  of  Section  II.  of 
Argelander's  "Zones,"  containing  105,075  stars. 

1862.  Section  III.  of  Argelander's  "Zones,"  containing  108,- 
129  stars.     Notes  on  989  Nebulse,  by  the  Earl  of  Rosse. 
Bond's  Memoir  on  Donati's  Comet  of  1858,  published 
in  the  Annals  of  the  Harvard  Observatory. 

1863.  Announcement  by  several  computers  that  the  accepted 
value  of  the  sun's  parallax  is  too  small  by  about  0".3. 
Spectroscopic    observations    of    celestial    objects,    by 
Huggins  and  Miller. 

1864.  Publication  of  Sir  John  Herschel's  great  catalogue  of 
5079  nebulae. 


APPENDIX  289 

A.D. 

1866.  Magnificent  display  of  shooting-stars  on  the  morning 
of  November   14th. 

1876.  First  use  of  dry  plates  in  celestial  photography. 

1877.  Hall  discovered  two  satellites  of  Mars. 

1881.  First  satisfactory  photograph  of  a  comet,  by  Tebutt. 
1885.  Photographic  star  charting  begun  by  Sir  David  Gill. 
1887.  Paris  catalogue  commenced.     Completed  in  1892. 
1891.  Hale    and    Deslandres    made    photographs    of    solar 

prominences. 
1900.  Cape    Photographic    Durchmunsturung,    cataloguing 

450,000  stars,  completed. 

NOTE. — The  third  edition  of  Chambers's  Descriptive  Astronomy 
(Oxford,  1877)  contains  an  exceedingly  interesting  and  elaborate  sum- 
mary of  Astronomical  Chronology.  A  similar  table  of  astronomical 
events  during  the  period  1774-1887  will  be  found  in  Clerke's  History 
of  Astronomy  during  the  Nineteenth  Century.  (Macmillan,  1887.) 


290  APPENDIX 


NAVIGATION 

THE  earliest  accounts  of  Navigation  appear  to  be  those 
of  Phoenicia,  1500  B.C.  Long  voyages  are  mentioned  in  the 
earliest  mythical  stories;  but  the  first  considerable  voyage 
of  even  probable  authenticity  seems  to  be  that  of  the  Phoe- 
nicians about  Africa,  600  B.C.  The  Roman  navy  dates  from 
311  B.C.,  and  that  of  the  Greeks  from  a  much  earlier  time. 
After  the  decay  of  these  nations,  commerce  passed  into  the 
hands  of  Genoa,  Venice,  and  the  Hanse  towns;  from  them  it 
passed  to  the  Portuguese  and  the  Spanish;  and  from  them 
again  to  the  English  and  the  Dutch. 

The  attractive  power  of  the  magnet  has  been  known  from 
time  immemorial;  but  its  property  of  pointing  to  the  north 
was  probably  discovered  by  Roger  Bacon  in  the  thirteenth 
century.  When  first  used  as  a  compass,  the  needle  was 
placed  upon  two  bits  of  wood,  which  floated  in  a  basin  of 
water;  and  the  method  of  suspending  it  dates  from  1302. 
The  variation  of  the  compass  was  discovered  by  Columbus, 
in  1492.  The  compass-box  and  the  hanging-compass  were 
invented  by  the  Rev.  William  Barlowe,  in  1608. 

Plane  charts  were  used  about  1420.  The  discovery  that 
the  loxodromic  curve  is  a  spiral  was  made  by  Nonius,  a 
Portuguese,  in  1537.  The  log  is  first  mentioned  by  Bourne, 
in  1577.  Logarithmic  tables  were  applied  to  Navigation  -by 
Gunter,  in  1620;  and  middle-latitude  sailing  was  introduced 
three  years  afterwards.  Other  dates  relating  to  the  subject 
of  Navigation  are  given  in  the  Astronomical  Chronology. 


APPENDIX 


291 


TABLE  I 


Name 

Symbol 

Incli- 
nation 
to  the 
Ecliptic 

Eccen- 
tricity of 
Orbit 

Greatest 
distance 
from 
Sun 

Least 
distance 
from 
Sun 

Mean 
distance 
from 
Sun 

Mean  dis- 
tance from 
Sun  in  miles 

Mercury 

fl 

7     0  12 

0.20562 

0.46817 

0  .  30603 

0.38710 

35,961,000 

Venus... 

9 

3  23  38 

0.00681 

0.72826 

0.71839 

0.72333 

67,197,000 

Earth... 

0or  £ 

0.01674 

1.01674 

0.98326 

1.00000 

92,897,416 

Mars.... 

3 

1  51  01 

0.09333 

1  .  66590 

1.38148 

1  .  52369 

141,551,000 

Jupiter.  . 

V 

1   18  28 

0.04837 

5.45446 

4.95114 

5  .  20280 

483,853,000 

Saturn.  . 

T? 

2  29  30 

0.05582 

10.07328 

9  .  00442 

9.53884 

887,098,000 

Uranus... 

Qor]£{ 

0  46  22 

0.04710 

20.07612 

18.28916 

19  .  19098 

1,784,732,000 

Neptune 

tJT 

1  46  39 

0.00855 

30.29888 

29.77506 

JO.  07067 

2,796,528,000 

Moon.  .  . 

* 

5  08  40 

0.05491 

Sidereal 

Daily 

Synodic 

Max. 

Min. 

Name 

Period  in 
tropical 

Helio- 
centric 

Period  in 
tropical 

Diameter 
from 

Diametei 
from 

at  mean 

Diameter 
in  miles 

years 

Motion 

years 

Earth 

Earth 

o       r    .  n 

/            n 

,             n 

/           // 

Mercury 

0.24085 

4  05  32 

0.31726 

12.9 

04.5 

6.6 

3009 

Venus.  .  . 

0.61521 

1   36  08 

1  .  59872 

1  06.3 

09.7 

16.  "9S 

7701 

Earth.... 

1.00004 

59  08 

• 

7926.7 

(equatorial) 

Mars.  .  .  . 

1.88089 

31  27 

2.13539 

30.1 

04.1 

9.6( 

)             4549 

Jupiter.  . 

11.86223 

4  59 

1.09211 

50.6 

30.  8 

I        37.  5( 

)          90256 

(equatorial) 

Saturn... 

29  .  45772 

2  00 

1.03518 

20.3 

14.  ( 

>         17.  5( 

)          76456 

Uranus... 

84.01529 

42 

1.01209 

04.3 

03.^ 

i          3.8( 

)          30193 

Neptune 

164  .  78829 

22 

1.00614 

02.9 

02.  ( 

)          2.2( 

)          34823 

days 

days 

Moon...  . 

27.322 

29.531 

33  31 

28  48 

31     07 

2159 

Sun  

32  35.6 

31  31 

32     03 

864392 

292 


APPENDIX 
TABLE  I — Continued 


Name 

Volume 
Earth  =  1 

Mass 
Sun=l 

Den- 
sity 
0=1 

Rotation 

Inclination 
of  Axis  to 
Ecliptic 

App. 
Diam.  of 
Sun  from 
Planet 

1~ 

d.    h.    m.    s  . 

/        // 

Mercury.  . 

0.052 

.633 

Not  known 

82  49 

10000000 

Venus  .... 

0.890 

1 

.913 

f 

Not  known 

44   19 

408000 

Earth  

1.000 

1 

1.000 

23  56  04 

0                   f 

66     33 

32  04 

333430 

Mars  

0.189 

1 

.666 

24  37  23 

64     51 

21  02 

3100000 

Jupiter.  .  . 

1390 

1 
1048 

.247 

9  55  41 

87     51 

6   10 

Saturn  .  .  . 

897.57 

1 
3500 

.123 

10  14  24 

65     28 

3  22 

Uranus  .  .  . 

55.25 

1 

.204 

10  (doubtful) 

1  40 

22900 

Neptune.  . 

84.85 

1 

.322 

' 

1  04 

19300 

Moon  

.020 

1 
27000000 

0.63 

27     7  43  11.5 

88     28 

Sun  

1,297,000 

1.0 

0.25 

25     9     07  12 

82     45 

Too  much  confidence  must  not  be  placed  in  the  absolute  accuracy  of 
all  the  elements  above  tabulated.  The  necessity  of  this  caution  will 
become  obvious  to  anyone  who  will  compare  the  values  of  these  elements 
as  they  are  given  by  different  authorities.  The  relative  distances,  the 
apparent  diameters,  and  the  periods  of  the  planets,  being  matters  of 
direct  observation,  are  known  to  a  great  degree  of  accuracy;  but  the 
absolute  distances  and  diameters,  and  the  volumes,  depending  as  they 
do  upon  the  distance  of  the  earth  from  the  sun,  must  be  considered  to 
be  only  approximately  known.  The  masses,  too,  of  some  of  the  planets 
are  uncertain :  and  so  also  must  be  the  densities,  which  depend  upon  the 
masses, 


APPENDIX  293 

TABLE  II 

THE    EARTH 


Density 5.67  (water  being  1) 

Polar  diameter 7899.98  statute  miles 

Equatorial  diameter 7926.68       "         " 

Oblateness 0.00337 

Length  of  sidereal  year 365d.  06h.  09m.  09.5s. 

"       "  tropical  year 365      05      48       45.9 

"       "anomalistic 365      06      13       53.1 

"       "  sidereal  day 23      56       04.09 

Eccentricity  of  orbit 0.0167431 

Inclination  of  orbit  in  1919 23°  26'  59".36 

Annual  diminution 0".4684 

"    precession 50".2568 

"    advance  of  line  of  apsides 11". 8 


TABLE  III 
THE   MOON 

Mean  distance  from  earth  in  earth's  radii 60.267 

" 238,862  statute  miles 

Greatest   "        "         " 251,978      " 

Least  " 225,746      " 

Sidereal  revolution 27d.     7h.  43m.   11.5s. 

Synodical       "       . . . : 29      12      44       02.8 

Mean  daily  geocentric  motion 13°  10'  36" 

Revolution  of  nodes 6793.43  days 

"  "  perigee 3232.58  days 

Mean  horizontal  parallax 57'  2".63 

Greatest        "  "     61    27 

Least  "  "     53    55 

Radius  in  terms  of  earth's  equatorial  radius 0.2725 

Mass      "       "      "     "      mass , -fa 

Density "       "      "     "      density f 


294 


APPENDIX 


TABLE  IV.— SATELLITES 

SATELLITES   OF   MARS 


Distance 

Sid. 

Names 

Inclination  of  orbit 
to  equator  of  primary 

from 
primary 

Period 

Dia. 
in 

Magni- 
tude 

in  miles 

d.     h.     m. 

miles 

1.  Phobos  

1°  42' 

5,800 

0  07  39 

36 

11 

2.  Deimos  

1     42 

14,650 

1  06   18 

11 

12 

SATELLITES   OF  JUPITER 


1.  lo  
2    Europa 

2°  08' 
1     39 
2     00 
1     57 
2     20 
30     00 
30     00 
148     00 
158     00  retrograde 

261,000 
415,000 
664,000 
1,167,000 
112,500 
7,208,000 
7,519,000 
16,031,000 

1   18  27 
3  13  13 
7  03  42 
16  16  32 
0  11  57 
251  00  00 
267  00  00 
1100  00  00 
745  00  00 

2400 
2100 
3500 
3000 
100 
31 
31 
20 
15 

6.0 
6.1 
5.6 
6.6 
13 
14    . 
14 
17 
19 

3.  Ganymede.  . 
4.  Calfisto  .... 
5  
6  

7  

8  
9  

SATELLITES   OF  SATURN 

1.   Mimas  
2.  Enceladus.  .  . 
3.  Tethys  
4.  Dione  

2°  18' 
2     18 
2     18 
2     18 
2     18 
1     57 
1     32 
7     14 
149     23  r 
13     24 

same  as 
plane  of 
rings 

etrograde 

117,000 
157,000 
186,000 
238,000 
332,000 
771,000 
934,000 
2,225,000 
8,000,000 
906,000 

0  22  37 
1  08  53 
1  21   18 
2   17  42 
4   12  25 
15  22  41 
'21  06  36 
79  07  56 
555   10  34 
20  20  24 

291 
368 
568 
540 
742 
1400 
192 
485 
42 
38 

17 
15 
12 
13 
10 
8 
17 
13 
18 
18 

5    Rhea 

6    Titan 

7.  Hyperion  .  .  . 
S.  lapetus  
9.  Phoebe  .  
10.  Themis  

SATELLITES   OF   URANUS 

1.  Ariel  
2.  Umbriel.  .  .  . 
3.  Titania  
4.  Oberon  

97°  58' 
98     21 
97     47 
97     54 

124,000 
173,000 
284,000 
380,000 

2   12  29 
4  03  28 
8  16  57 
13   11  07 

360 
250 
800 
744 

16 
16 
14 
14 

SATELLITE   OF   NEPTUNE 

1 

142°  40' 

220,000 

5  21  03 

1950 

14 

APPENDIX 


295 


TABLE  V 
THE  MINOR  PLANETS 


No. 

Name 

Year 
of  Dis- 
cov- 
ery 

Discoverer 

Inclina- 
tion 

Eccen- 
tricity 

Period 
in 
Years 

Distance 
in  Radii 
of 
Earth's 
orbit 

Diam- 
eter in 
miles 

1 

Ceres  

1801 

Piazzi  

10°36' 

0.080 

4.6 

2.77 

227 

2 

Pallas  

1802 

Gibers  

34  42 

0.240 

4.6 

2.77 

172 

3 

Juno  

1804 

Harding  

13  03 

0.256 

4.4 

2.67 

112 

4 

Vesta  

1807 

Gibers  

7  08 

0.090 

3.6 

2.36 

228 

5 

Astraea  

1845 

Hencke  

5   19 

0.190 

4.1 

2.58 

61 

6 

Hebe  

1847 

Hencke  

14  46 

0.201 

3.8 

2.43 

100 

7 

Iris.  .  

Hind  

5  27 

0.231 

3.7 

2.39 

96 

8 

Flora  

Hind  

5  53 

0.157 

3.3 

2.20 

60 

9 

Metis  

1848 

Graham  

5  36 

0.123 

3.7 

2.39 

76 

10 

Hygeia  

1849 

De  Gasparis. 

3  47 

0.120 

5.6 

3.14 

111 

11 

Parthenope  . 

1850 

De  Gasparis  . 

4  36 

0.099 

3.8 

2.45 

62 

12 

Victoria  .... 

Hind  

8  23 

0.219 

3.5 

2.33 

41 

13 

Egeria  

De  Gasparis. 

16  32 

0.088 

4.1 

2.58 

73 

14 

Irene  

1851 

Hind  .  .  :  

9  07 

0.165 

4.2 

2.59 

68 

15 

Eunomia.  .  . 

De  Gasparis  . 

11  44 

0.188 

4.3 

2.64 

12 

16 

Psyche  

1852 

De  Gasparis.  . 

3  04 

0.136 

5.0 

2.86 

93 

17 

Thetis  

Luther  

5  35 

0.133 

3.9 

2.47 

52 

18 

Melpomene  . 

Hind  

10  09 

0.217 

3.5 

2.30 

54 

19 

Fortuna  .... 

Hind 

1   32 

0.  158 

3.8 

2.44 

61 

20 

Massilia.  .  .  . 

De  Gasparis.  . 

0  41 

0.144 

3.7 

2.41 

68 

21 

Lutetia  

Goldschmidt. 

3  05 

0.162 

3.8 

2.44 

40 

22 

Calliope.  .  .  . 

Hind  

13  44 

0.098 

5.0 

2.91 

96 

23 

Thalia  

Hind  

10  13 

0.232 

4.3 

2.62 

42 

24 

Themis  

1853 

De  Gasparis.  . 

0  48 

0.137 

5.6 

3.14 

36 

25 

Phocea  

Chacornac.  .  . 

21   34 

0.253 

3.7 

2.40 

31 

26 

Proserpine  .  . 

Luther  

3  35 

0.088 

4.3 

2.66 

47 

27 

Euterpe  .... 

Hind  

1   35 

0.173 

3.6 

2.35 

39 

28 

Bellona  

1854 

Luther  

9  21 

0.150 

4.6 

2.78 

59 

29 

Amphitrite.  . 

Marth  

6  07 

0.072 

4.1 

2.55 

83 

30 

Urania  

Hind  

2  05 

0.127 

3.6 

2.36 

51 

31 

Euphrosyne 

Ferguson.  .  .  . 

26  25 

0.223 

5.6 

3.16 

50 

32 

Pomona.  .  .  . 

Goldschmidt. 

5  29 

0.082 

4.2 

2.58 

35 

33 

Polyhymnia. 

Chacornac.  .  . 

1  56 

0.338 

4.8 

2.86 

38 

34 

Circe  

1855 

Chacornac.  .  . 

5  26 

0.110 

4.4 

2.68 

29 

35 

Leucothea.  . 

Luther  

8  05 

0.223 

5.2 

3.01 

25 

36 

Atalanta.  .  .  . 

Goldschmidt. 

18  42 

0.298 

4.6 

2.75 

20 

37 

Fides 

Luther  

3  07 

0.  175 

4.3 

2.64 

41 

38 

Leda  

1856 

3hacornac.  .  . 

6  58 

0.  172 

4.5 

2.74 

29 

39 

Lsetitia  

Chacornac  .  .  . 

10  21 

0,111 

4.6 

2.77 

87 

40 

Harmonia 

Luther  

4  15 

0.046 

3.4 

2.27 

296 


APPENDIX 
TABLE  V— Continued 


No, 

Name 

Year 
of  Dis- 
cov- 
ery 

Discoverer 

Inclina- 
tion 

Eccen- 
tricity 

Period 
in 
Years 

Distance 
in  Radii 
of 
Earth's 
orbit 

Diam- 
eter in 
miles 

41 
42 
43 
44 

Daphne  .... 
Isis  
Ariadne.  .  .  . 
Nysa  

1857 

Goldschmidt. 
Pogson  
Pogson  
Goldschmidt  . 

15°55' 
8  35 
3  27 
3  41 

0.270 
0.226 
0.168 
0.149 

4.6 
3.8 
3.3 
3.8 

2  77 
2.44 
2.20 
2.42 

45 

Eugenia.  .  .  . 

Goldschmidt. 

6  34 

0.082 

4.5 

2.72 

46 

Hestia  

Pogson  

2   17 

0.162 

4.0 

2.52 

47 

48 

Melete  
Aglaia 

1857 

Goldschmidt. 
Luther 

8  01 
5  00 

0.237 
0   128 

4.2 
4  9 

2.60 
2  88 

49 

Doris 

Goldschmidt  . 

6  29 

C  061 

5  5 

3  10 

50 

Pales      . 

Goldschmidt  . 

3  08 

0.238 

5  4 

3  09 

51 
52 
53 
54 
55 
56 

Virginia  .... 
Nemausa.  .  . 
Europa  
Calypso  .... 
Alexandra.  . 
Pandora 

1858 

Ferguson.  .  .  . 
Laurent  
Goldschmidt. 
Luther  
Goldschmidt. 
Searle. 

2  47 
9  57 
7  24 
5  07 
11  47 
7  20 

0.287 
0.063 
0.111 
0.204 
0.199 
0   139 

4.3 
3.7 
5.5 
4.2 
4.6 
4  6 

2.65 
2.38 
3.10 
2.61 
2.71 
2  77 

57 

1859 

Luther 

15  12 

0   108 

5  6 

3.1fi 

58 
59 

Concordia.  . 
Danae  .  .    . 

1860 

Luther  
Goldschmidt  . 

5  02 
18  17 

0.041 
0.163 

4.4 
5.1 

2.70 
2.97 

60 
61 

Oly  mpia  .... 
Erato  

Chacornac.  .  . 
Forster  

8  36 
2   12 

0.119 
0.185 

4.5 
5.5 

2.71 
3.13 

62 
63 
64 

Echo  
Ausonia  .... 
Angelina.  .  .  . 

1861 

Ferguson.  .  .  . 
De  Gasparis.  . 
Tempel  

3  34 
5  45 
1   19 

0.185 
0.127 
0.125 

3.7 
3.7 
4.4 

2.39 
2.40 
2.68 

65 

Cybele  

Tempel  

3  28 

0.100 

6.3 

3.42 

66 

Maia 

Tuttle 

3  04 

0.174 

4.3 

2.65 

67 

Asia 

Pogson  .  .    .  . 

5  59 

0.184 

3.8 

2.42 

68 
69 

Hesperia  
Leto 

Schiaparelli.  . 
Luther  

8  28 

7  58 

0.175 
0.186 

5.2 
4.6 

2.99 
2.77 

70 
71 

Panopea.  .  .  . 
Feronia  

Goldschmidt. 
C.H.F.  Peters 

11  39 
5  24 

0.183 
0  120 

4.2 
3.4 

2.61 
2.27 

72 

Niobe  

Luther  

23   19 

0.174 

4.6 

2.76 

73 

Clytie 

1862 

Tuttle     .  .  . 

2  25 

0.044 

4.3 

2.66 

74 

Galatea 

Tempel 

3  59 

0.238 

4.6 

2.78 

75 
76 

Eurydice.  .  . 
Freia 

Peters  
D'  Arrest.  .  .  . 

5  00 
2  02 

0.307 
0.173 

4.4 
6.2 

2.67 
3.39 

77 
78 
79 
80 

Frigga  
Diana.  ..... 
Eurynome.  . 

1863 
1864 

Peters  
Luther  
Watson  
Pogson  .  . 

2  28 
8  39 
4  37 
8  37 

0.136 
0.205 
0.195 
0.200 

4.4 
4.2 
3.8 
3.5 

2.67 
2.62 
2.44 
2.30 

81 

82 

Terpsichore  . 

Tempe  
Luther  

7  56 
2  51 

0.212 
0.226 

4.8 
4.6 

2.86 
2.76 

83 

Beatrix  

1865 

De  Gasparis.  . 

5  02 

0.084 

3.8 

2.43 

APPENDIX 


297 


TABLE  \-Continued 


Voor 

Distance 

No. 

Name 

of  Dis- 
cov- 

Discoverer 

Inclina- 
tion 

Eccen- 
tricity 

Period 
in 
Years 

in  Radii 
of 
Earth's 

Diam- 
eter in 
miles 

ery 

orbit 

84 

Clio 

Luther  

9C22' 

0.238 

3   6 

2  37 

85 

lo           ... 

Peters  

11   56 

0   194 

4.3 

2.66 

86 

Semele  

1866 

Tietjen  

4  48 

0.213 

5.5 

3.11 

87 

Sylvia     .... 

Pogson  

10  53 

0.095 

6  5 

3.49 

88 

Thisbe  

Peters  

5  09 

0  Ib7 

4.6 

2.77 

89 

Julia  

Stephan  

16  07 

0.183 

41 

2.55 

90 

Antiope.  .  .  . 

Luther  

2   15 

0.148 

5.6 

3.16 

91 

wjEgina  

St'phan  

2     8 

0   106 

4.2 

2.58 

The  number  of  minor  planets  discovered  up  to  1919  was  over  800.  The  diam- 
eters are  derived  from  photometric  experiments  made  by  Professor  Stampfer,  of 
Vienna.  They  are  probably  only  relatively  correct. 

TABLE  VI 
SCHWABE'S  OBSERVATIONS  OF  THE  SOLAR  SPOTS 


Year. 

Number 
of  days 
of  obser- 
vation 

New  Groups 

Days  on 
which 
the  Sun 
was  free 
from 
spots 

Year 

Number 
of  days 
of  obser- 
vation 

New  Groups 

Days  on 
which 
the  Sun 
was  free 
from 
spots 

1826 

277 

118 

22 

1846 

314 

157 

1 

7 

273 

161 

2 

7 

276 

257 

0 

8 

282 

225  (Max.) 

0 

8 

278 

330(Max.) 

0 

9 

244 

199 

0 

9 

285 

238 

0 

1830 

217 

190 

1 

1850 

308 

186 

2 

1 

239 

149 

3 

1 

308 

151 

0 

2 

270 

84 

49 

2 

337 

125 

2 

3 

247 

33  (Min.) 

139 

3 

299 

91 

3 

4 

273 

51 

120 

4 

334 

67 

65 

5 

244 

173 

18 

5 

313 

79 

146 

6 

200 

272 

0 

6 

321 

34  (Min.) 

193 

7 

168 

333  (Max.) 

0 

7 

324 

98 

52 

8 

202 

282 

0 

8 

335 

188 

0 

9 

205 

162 

0 

9 

343 

205 

0 

1840 

263 

152 

3 

1860 

332 

211  (Max.) 

0 

1 

283 

102 

15 

1 

322 

204 

0 

2 

307 

68 

64 

2 

317 

160 

3 

3 

312 

34  (Min.) 

149 

3 

330 

124 

2 

4 

321 

52 

111 

4 

325 

130 

4 

1845 

332 

114 

29 

1865 

307 

93 

25 

298  APPENDIX 

TABLE  VII 

TRANSITS    OF   THE    INFERIOR    PLANETS 


Mercury. 

Venus. 

1802. 

November    8. 

1639. 

December    4. 

1815. 

November  11. 

1761. 

June             5. 

1822. 

November    4. 

1769. 

June             3. 

1832. 

May              5. 

1874. 

December    8. 

1835. 

November    7. 

1882. 

December    6. 

1845. 

May              8. 

2004. 

June             7. 

1848. 

November    9. 

2012. 

June             5. 

1861. 

November  11. 

2117. 

December  10. 

1868. 

November    4. 

2125. 

December    8. 

1878. 

May              6. 

2247. 

June           11. 

1881. 

November    7. 

2255. 

June             8. 

1891. 

May              9. 

2360. 

December  12. 

1894. 

November  10. 

2368. 

December  10. 

1907. 

November  13. 

1914. 

November  6-7. 

APPENDIX 


299 


TABLE  VIII 

THE    PRINCIPAL    CONSTELLATIONS 
Those  found  in  Ptolemy's  Catalogue  (137  A.D.)  are  in  Italics 

THE    NORTHERN    CONSTELLATIONS 


No. 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 

Name. 

Co-ordinates  of 
Center. 

Name  of 
Principal  Star  of 
1st  or  2d 
magnitude. 

Number 
of  stars 
of  1st 
mag. 

Number 
of  stars 
of  first 
five  mag- 
nitudes. 

R.  A. 

D. 

Andromeda  
Cassiopeia  
Triangulum  
Perseus  
Camelopardus  
Auriga  
Lynx  

h.   m. 
1        0 
1      10 
2       0 
3     30 
6     45 
6       0 
7     55 
10     05 
10     40 
12     40 
13        0 
14     35 
15       0 
15     40 
15     40 
17      10 
17     20 
17     50 
18     40 
19     30 
19     40 
20       0 
20     20 
20     40 
21        0 
21     40' 
22     20 
22     25 

0 

35 
60 
32 
47 
68 
42 
50 
36 
58 
26 
40 
30 
78 
30 
10 
27 
66 
5 
35 
10 
18 
25 
42 
15 
6 
65 
44 
15 

Alpheratz  (a) 

Mirfak  (a) 
Capella  («) 

Dubhe  (a) 

Cor  Caroli  (a) 
Arcturus  (a) 
Polaris  (a) 

Unukalhay  («) 
Rasalgeu  (a) 
Thuban  (a) 

Vega  («) 
Altair  (a) 

Deneb  (a) 
Markab  (a) 

1 

1 
1 

1 

1 
1 

18 
46 
5 
40 
36 
35 
28 
15 
53 
20 
15 
35 
23 
19 
23 
65 
80 
6 
18 
33 
5 
23 
67 
10 
.5 
44 
13 
43 

Leo  Minor  
Ursa  Major  
Coma  Berenicis.  .  .  . 
Canes  Venatici  
Bootes 

Corona  Borealis  
Serpcns  
Hercules 

Draco 

Taurus  Poniatowskii 
Lyra  . 

Aquila  
Sagitta  
Vulpecula  et  Anser. 
Cygnus  
Delphinus  

Equuleus  

Cepheus  

Lacerta  

Pegasus  

28 

Total  

6 

823 

300 


APPENDIX 


TABLE  VIII— Continued 
THE  ZODIACAL  CONSTELLATIONS. 


No. 

Name. 

Co-ordinates  of 
Center. 

Name  of 
Principal  Star 
of  1st  or  2d 
magnitude. 

Number 
of  stars 
of  1st 
mag. 

Number 
of  stars 
of  first 
five  mag- 
nitudes. 

R.     A. 

D. 

1 

2 

3 

4 
5 
6 
7 
8 
9 
10 
11 
12 

Aries  

h.     m. 
2     30 
4       0 

7       0 

8     40 
10     20 
13     20 
15       0 
16     15 
18     55 
21       0 
22        0 
0     20 

0 

18  N 
18 

25 

20 
15 
3  N 
15  S 
26 
32 
20 
9  S 
ION 

Hamal  (a) 
Aldebaran  (a) 
f    Castor  (a) 
\    Pollux  (0) 

Regulus  (a)  
Spica  (a) 
Zuben-el-gubi  (a; 
Antares  (a) 

SecundaGiedi  (a; 
Sadalmelik  (a) 

1 
1 

1 
1 

1 

17 
58 

28 

15 

47 
39 
23 
34 
38 
22 
25 
18 

Taurus  
Gemini  

Cancer  

Leo 

Virgo 

Libra 

Scorpio  .  .  . 

Sagittarius  

Capricornus  
Aquarius  
Pisces  

12 

Total    . 

5 

364 

THE  SOUTHERN  CONSTELLATIONS. 


1 

Apparatus  Sculptoris 

0     20 

32 

13 

2 

Phoenix  

1       0 

50 

32 

3 

Cetus  

2       0 

12 

Menkar  (a) 

32 

4 

Fornax  Chemica.  .  .  . 

2     20 

30 

6 

5 

Hydrus  

2     40 

70 

25 

6 

Horologium  

3     15 

57 

11 

7 

Eridanus  

3     40 

30 

Achernar  (a) 

1 

64 

8 

Reticulus  Rhomboi- 

dalis.      . 

4       0 

62 

11 

9 

Dorado  

4     40 

62 

17 

10 

Caela  Sculptoria  .... 

4     40 

42      • 

6 

11 

Mons  Mens  •>  

5     20 

75 

9 

12 

Columbia  Noachi.  .  . 

5     25 

35 

Phact  (a) 

15 

13 

Equuleus  Pictoris.  .  . 

5     25 

55 

17 

14 

Lepus  , 

5     25 

20 

Arneb  (a) 

18 

15 

Orion  

5     30 

0 

Rigel  (0) 

2 

37 

16 

Canis  Major  

6     45 

24 

Sirius  (a) 

1 

27 

17 

Monoceros  

7       0 

2 

12 

18 

Canis  Minor  

7     25 

5 

Procyon  (a) 

1 

6 

19 

A.TQO 

7     40 

50 

Canopus  (a) 

2 

133 

20 

Piscis  Volans  

7     40 

68 

9 

21 

Sextans  ...    

10       0 

0 

3 

APPENDIX 


301 


TABLE  VIII— Continued. 
THE   SOUTHERN    CONSTELLATIONS 


No. 

Name. 

Co-ordinates  of 
Center. 

Name  of 
Principal  Star 
of  1st  or  2d 
magnitude. 

Number 
of  stars 
of  1st 
mag. 

Number 
of  stars 
of  first 
five  mag- 
nitudes. 

R.     A. 

D. 

22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 

Hydra  

h.     m. 
10       0 
10       0 
10     50 
11     20 
12     15 
12     20 
12     25 
13       0 
15       0 
15     20 
15     25 
15     40 
16       0 
17       0 
17       0 
18     10 
18     30 
18     40 
19     20 
20     40 
21       0 
21       0 
21     40 
22     20 
23     45 

0 

10 
35 
78 
15 
60 
18 
68 
48 
64 
76 
45 
65 
45 
54 
0 
15  8 
40 
53 
68 
37 
55 
80 
32 
47 
66 

Alphard  (a) 
Algorab  (a) 

R  as  a  1  ague  (a) 

Formalhaut  (a) 

• 

• 
1 
2 

1 

49 
7 
17 
9 
10 
8 
7 
54 
2 
7 
34 
11 
12 
15 
40 
4 
7 
8 
27 
5 
15 
16 
16 
11 
21 

Antila  Pneumatica^  . 
Chamseleon  

Crater 

Crux 

Corcus  
Musca  Australis.  .  .  . 
Centaurus  
Circinus 

A  pus 

Lupus  .  .  . 

Triangulum  Australe 
Norma.    ... 

Ara  .    . 

Ophiuchus.  . 

Scutum  Sobieskii  .  .  . 
Corona  Australis  
Telescopium  
Pavo 

Microscopium  
Indus  
Octans  
Piscis  Australis  
Grus  

Toucan  

46 

Total  

11 

915 

86 

Summary  

22 

2102 

302 


APPENDIX 


TABLE  IX 

VARIABLE    STARS 


Name. 

Co-ordinates,  1919 

Period  in 
Days. 

Change  of 
Magni- 
tude. 

Authority. 

R.  A 

D. 

a  Cassiopeia  . 
o  Ceti  
p  Persei  
0  Persei  
X  Tauri 

h.   m.   s. 
0  35  54 
2   15   15 
2  59   59 
3  02   54 
3  56   11 
4  56  09 
5  50  47 
6  09   59 
6  59   18 
10  41   55 
13  57  23 
14   56  38 
17   10  57 
18  47  05 
18  52  52 
19  48  21 
22  59   51 

58  05.6  N 
3  20.7    S 
38  31.6  N 
40  38.7  N 
12   15.7  N 
43  42.3  N 
7  23.6  N 
22  31.9  N 
20  41.4  N 
59   15.3    S 
76  24.4    S 
8  11.9    S 
14  28.9  N 
33  16.1  N 
43  50.3  N 
0  47.8  N 
27  38.6  N 

Irregular 
331. 
Irregular 
2.87 
3.95 
Irregular 
Irregular 
231.4 
10.15 
Irregular 
Irregular 
2.33 
Irregular 
12.9 
46.4 
7.18 
Irregular 

2  .  2  to  2  .  8 
1.7   "  9.6 
3.4   "  4.2 
2.1   "  3.2 
3.3   "  4.2 
3.0   "  4.5 
1.0  "   1.4 
3.2   "  4.2 
3.7   "  4.3 
1.6  "  6.6 
5.5  "  6.6 
4.8  "  6.2 
3.1   "  3.9 
3.4  "  4.1 
4.0  "  4.7 
3.7  "  4.4 
2.2  "  2.7 

Birt,                  1831 
D.  Fabricius,  1596 
Schmidt,           1854 
Montanari,      1669 
Baxendell,       1848 
Heis,                 1846 
J.  Herschel,     1840 
Schmidt           1865 
Schmidt,          1847 
Burchell,          1827 
Naut.  Aim. 
Schmidt           1859 
W.  Herschel    1795 
Gcodricke,      1784 
Baxendellt       1856 
Pigott,              1784 
Schmidt,          1847 

c  Aurige  
a  Orionis  
tj  Geminorum 
f  Geminorum 
i?  Argus  
e  Apodis  
6  Librae  
a  Herculis.  .  .  . 
/3  Lyrae.. 

RLyrae  
n  Aquilse  
0  Pegasi  

NOTE. — Only  those  variable  stars  listed  in  the   Nautical  Almanac  are  listed  in 
this  table, 


APPENDIX 


303 


TABLE  X. 

BINARY   STARS 


Co-ordinates, 

Name 

1920 

Magnitude 
of  Com- 
ponents 

Spec- 
trum 

Paral- 
lax 

Period 
in  Years 

Semi- 
major 
Axis 

Com- 
bined 

Mass 

R.  A. 

D. 

h.     m. 

0          / 

n 

» 

a  Centauri.  . 

14  34 

60  30  S 

0.3   1.7 

G    0 

0.76 

81.2 

15.50 

1.9 

a  Canis 

Majoris  . 

6  42 

16  36   S 

—  1.6  5.0 

A    0 

0.38 

50.0 

7.0 

3.4 

a  Canis 

Minoris 

7  35 

5  26  N 

0.5  4.5 

F    5 

0.32 

39.0 

3.0 

0.7 

t}  Cassiopeia. 

0  44 

7  09  N 

3.6  7.6 

F   8 

0.20 

300  (?) 

40 

1.2 

70  Ophiuchi  . 

18  01 

2  31  N 

4.1  6.0 

K    0 

0.17 

88.4 

5.5 

2.5 

6  Eridani.  .  . 

2  55 

40  37   S 

3.4  4.4 

A    2 

0.17 

180 

8.0 

0.7 

72  Pegasi.... 

23  30 

30  53  N 

6.0  6.0 

K   2 

0.11 

26.3 

0.4 

0,7 

f  Herculis.  .  . 

16  38 

31  45  N 

3.0  6.0 

G    0 

0.10 

34  5 

1.0 

2.1 

n  Bootis.  .  .  . 

15  21 

37  39  N 

4.5  6.7 

F    0 

0.05 

650 

3.21 

0.2 

NOTE. — There  are  many  thousands  of  other  binaries. 

The  spectral   notation  is  that  of   the  Annals  of  Harvard   University,    Vol.  LVI. 
The  mass  of  the  binaries  is  given  in  terms  of  the  sun's  mass. 


PLATE  I. 


COMPARATIVE  SIZES  OF  THE  PLANETS. 

1.  MERCURY.  5.  URANUS. 

2.  MARS.  6.  NEPTUNE. 

3.  VENUS.  7.  SATURN. 

4.  EARTH.  8.  JUPITER. 


PLATE  II. 


EQUATORIAL. 

The  40-inch  Refractor  of  the  Yerkes  Observatory, 
Williams  Bay,  Wisconsin. 


PLATE  III. 


Photograph  bv  Messrs.  Barnard  and  Rltchey,  Yerkes  Observatory. 

TOTAL  ECLIPSE  OF  SUN,  MAY  28,  1900. 


PLATE  IV. 


Photographed  by  Professor  E.  E.  Barnard,  Yerkes  Observatory. 

COMET  MOREHOUSE,  1908,  NOVEMBER  16. 

This  shows  the  transparency  of  a  comet's  tail.    Note  also  the  short 
streams  of  matter  ejected  at  a  considerable  angle  with  the  main  tail. 


PLATE  V. 


Photographed  by  Barnard.  YerTces  Observatory. 


COMET  BELJAWSKY,  1911,  SEPEMBEB  29. 

Shows  the  head  of  a  large  comet  and  the  envelopes  that  surround  it 
and  stream  back  to  form  the  tail. 


PLATE  VI. 


Photographed  by  Barnard,  Yerkes  Observatory. 

COMET  BROOKS,  1911,  OCTOBER  19. 

The  definite  outlines  of  the  tail  near  the  head  are  remarkable.   Much 
structure  is  also  shown  in  the  tail. 


PLATE  VII. 


Photographed  bv  Barnard,  Yertes  Observatory. 


HALLEY'S  COMET,  1910,  MAT  29. 

Very  nearly  the  naked  eye  appearance  of  a  large  comet.     The  tail  is 
soft  and  diffuse  with  very  little  structure. 


PLATE  VIII. 


Photographed  at  Yerkes  Observatory. 

THE  GREAT  NEBULA  IN  ANDROMEDA. 


PLATE  IX. 


Photographed  with  the  Crosslev  Reflector  of  the  Lick  Observatory. 

THE  GREAT  SPIRAL  NEBULA  M-33,  IN  TRIANGULUM 


INDEX 


(The  references  are  to  the  pages.) 


Aberration  of  light,  124;  diurnal, 
annual,  126;  separated  from  par- 
allax, 249;  planetary,  128.* 

Acceleration,  secular  of  the  moon, 
144. 

Aerolites,  235. 

Algol,  253. 

Altazimuth,  34. 

Altitude,  10. 

Altitude  and  azimuth  instrument, 
33;  use  of,  35. 

Altitudes,  method  of  equal,  34. 

Amplitude,  10. 

Andromeda,  nebula  in,  262. 

Annular  eclipse,  154. 

Anomaly,  95. 

Aphelion,  95. 

Apogee,  131. 

Appulse,  148. 

Apsides,  of  earth,  124;  of  moon, 
131;  of  planets,  164. 

Aquilae,  nova,  254. 

Arc  of  meridian  measured,  61. 

Argo,  nebula  in,  264. 

Aries,  first  point  of,  12,  87. 

Ascension,  right,  12;  related  to 
sidereal  time,  14. 

Asteroids,  189;  table  of,  295. 

Astronomy,  1;  chronology  of,  282. 

Atmosphere,  height  of,  51. 

Attraction,  law  of,  114. 

Axis,  of  the  heavens,  8;  of  the 
earth,  8;  of  collimation,  27. 

Azimuth,  10. 


Base-line,  61. 
Bode's  law,  189. 
Branches  of  meridian,  9. 

Calendar,  112. 

Centauri,  alpha,  distance  of,  250; 
a  binary  star,  256. 

Centrifugal  force,  67,  280. 

Ceres,  discovery  of,  190. 

Chromosphere,  100. 

Chronograph,  23. 

Chronometer,  Greenwich  time 
given  by,  80. 

Circle,  vertical,  10;  hour,  11;  of 
perpetual  apparition,  11;  diur- 
nal, 16;  of  latitude,  87;  vanish- 
ing, 18. 

Circle,  meridian,  28;  reflecting,  47. 

Clock,  astronomical,  21;  driving, 
37. 

Clusters  of  stars,  260. 

Coal  sack,  265. 

Collimation,  axis  of,  27. 

Colures,  87. 

Comets,  208;  diversity  of  appear- 
ance, 209;  tail,  210;  orbits,  212; 
periods  and  motion,  214;  mass 
and  density,  215;  light,  216; 
periodic,  218;  Encke's,  218; 
Winnecke's  or  Pons's,  220;  Bror- 
sen's,  202;  Biela's,  221;  D' Ar- 
rest's, 222;  Faye's,  222;  Me- 
chain's,222;  Halley's,223;  Great 
of  1811,  224;  of  1843,  224; 


305 


306 


INDEX 


Donating,    225;    of    1861,  '226; 

connection  with  meteors,   237; 

list  of  periodic,  217. 
Compression,  65. 
Conjunction,     136;    inferior    and 

superior,  174. 

Constellations,  243;  list  of,  299. 
Co-ordinates,  spherical,  17. 
Corona,  100. 
Count,  least,  46. 
Crescent  moon,  137. 
Cross-wires,  27. 
Culmination,  11. 

Cycle,  lunar,  144;  of  eclipses,  156. 
Cygni,  61 ;    distance  of,  250 

Day,  solar  and  sidereal,  106;  in- 
equality of  solar,  107;  astro- 
nomical and  civil,  111;  intercal- 
ary, 112. 

Declination,  11. 

Degree  of  meridian,  63. 

Departure,  9. 

Dip  of  horizon,  57. 

Dipper,  243. 

Disc,  spurious,  of  stars,  251. 

Distance,  zenith,  10;  polar,  11. 

Earth,  general  form  of,  2;  spheroi- 
dal form,  63;  dimensions,  64; 
density,  65;  linear  velocity  of 
rotation,  73;  revolution  about 
the  sun,  92;  orbital  velocity,  93; 
orbit,  98;  motion  at  perihelion 
and  aphelion,  117;  phases,  138; 
elements,  293. 

Eccentricity  of  an  ellipse,  95. 

Eclipses,  147;  lunar,  147;  solar, 
152;  total,  155;  cycle  of,  156; 
number  of,  157;  of  Jupiter's 
satellites,  193. 

Ecliptic,  86. 


Ecliptic  limits,  lunar,  150;  solar, 

154. 
Elements,  of  planetary  orbit,  176; 

of  cometary,  212. 
Ellipse,  278. 
Elongation,    56;   greatest  eastern 

and  western,   174. 
Equation,  of  center,  109;  of  time, 

111;  annual,    144. 
Equator,  8;  celestial,  10. 
Equatorial,  36. 
Equilibrium    of    centrifugal    and 

centripetal  forces,  114. 
Equinoctial,  10. 
Equinox,  vernal,  12,  86. 
Error  of  clock,  22. 
Errors  of  observation,  50.     , 
Establishment  of  port,  166. 
Evection,  143. 
Evening  star,  176,  188. 

Faculae,  102. 
Finder,  28. 
Flames,  red,  102. 

Forces,    centrifugal    and    centri- 
petal, 281. 
Foucault's  experiment,  71. 

Galaxy,  265. 
Gemini,  244. 
Geocentric,  parallax,  54;  motion  of 

planets,  171,  174,  184. 
Gibbous,  137. 
Golden  number,  145. 
Granulations,  102. 
Gravitation,  universal,  114. 

Heliocentric,  parallax,  54,  171; 
'motion  of  planets,  178,  185. 

Hemisphere,  8. 

Horizon,  3;  points  of,  4;  artifi- 
cial, 44 

Horizontal  point,  31. 


INDEX 


307 


Hour  angle,  12;  relation  of  sider- 
eal time,  14. 
Hour  circle,  11. 
Hyades,  261. 
Hyperbola,  279. 

Incidence,  angle  of,  50. 
Index  correction,  43. 

Jupiter,  191;  mass  of,  195. 
Kepler's  laws,  118. 

Latitude,  9,  74;  equal  to  altitude 
of  pole,  14;  methods  of  deter- 
mining, 74;  at  sea,  77;  reduction 
of,  78;  celestial,  87. 

Level,  hanging,  28. 

Librations,  142. 

Light,  analysis  of,  50;  of  sun,  102; 
velocity  of,  126;  of  planets,  203; 
of  stars,  246;  of  nebulae,  260; 
of  comets,  216;  zodiacal,  104. 

Line  of  sight,  27. 

Longitude,  9,  79;  how  determined, 
79;  by  telegraph,  85;  by  star 
signals,  82;  at  sea,  83;  celestial, 
87;  by  eclipses  and  occulta- 
tions,  159. 

Lunar  distance,  81. 

Lunation,  138. 

Magellanic  clouds,  263. 

Magnitudes,  241. 

Mars,  188. 

Mercury,  181;  transits  of,  298. 

Meridian,  terrestrial,  8;  prime,  9; 

celestial,  9;  line,  10. 
Meteors,  227;  showers,  228;  height 

and  velocity,  231;  orbits,  231; 

detonating,  234. 
Micrometer,  37. 


Microscope,  reading,  29. 

Milky  way,  265. 

Minor  planets,  189;  list  of,  295. 

Mira,  or  0  Ceti,  252. 

Moon,  129;  nodes,  130;  obliquity 
of  orbit,  131;  line  of  apsides, 
131;  meridian  zenith  distance, 
132;  distance,  134;  size  and 
mass,  135;  augmentation  of  di- 
ameter, 135;  phases,  136;  sider- 
eal and  synodic  periods,  138; 
retardation,  140;  harvest,  141; 
rotation,  141;  librations,  142; 
general  description,  145;  ele- 
ments, 293. 

Morning  star,  176,  188. 

Motion,  diurnal,  3;  upward  and 
downward,  9;  west  to  east,  92 
(note);  direct  and  retrograde, 
174,  184. 

Nadir,  9;  point,  31,  32. 

Navigation,  sketch  of,  290. 

Nebulae,  259;  annular,  261;  el- 
liptic, 262;  spiral  and  planet- 
ary, 262;  nebulous  stars,  263; 
double,  263;  variation  of  bright- 
ness, 264. 

Nebular  hypothesis,  204. 

Neptune,  202;  immense  distance 
of,  203. 

Nodes,  of  moon's  orbit,  130; 
heliocentric  longitude  of 
planet's,  177.  . 

Noon,  11. 

Nubeculae,  263. 

Nutation,  123. 

Obliquity  of  ecliptic,  109,  124. 
Occultation,  147,  157;  of  Jupiter's 

satellites,  194. 
Octant,  47. 


308 


INDEX 


Olbers's  theory,  191. 
Opposition,  136.. 
Orion,  244. 

Parabola,  279. 

Parallax,  54;  geocentric  and  hori- 
zontal, 55;  heliocentric,  56,  171; 
annual,  247. 

Pegasus,  244. 

Pendulum  experiment,  71. 

Penumbra,  of  solar  spots,  101;  of 
eclipse,  148,  152. 

Perigee,  131, 

Perihelion,  95. 

Perturbations,  in  earth's  orbit, 
120;  in  moon's,  141. 

Phases,  of  moon,  15;  earth,  138; 
Mercury  and  Venus,  182;  Mars, 
188.  . 

Photography,  47. 

Photosphere,  100. 

Planetoids,  189;  list  of,  295. 

Planets,  171;  orbits  of,  173;  in- 
ferior, 174;  stationary  points, 
175,  186;  elements  of  orbit,  176; 
heliocentric  longitude  of  node, 
177;  inclination  of  orbit,  179; 
periods,  180,  186;  superior,  184; 
distance,  187;  elements,  291; 
satellites  of,  294. 

Plateau's  experiment,  206. 

Pleiades,  260. 

Points,  fixed,  31. 

Pointers,  243. 

Poles,  of  the  heavens,  8;  of  the 
earth,  8. 

Pole-star,  5,  123. 

Position  angle,  15. 

Praesepe,  261. 

Precession,  120. 

Problem  of  three  bodies,  144. 

Projections,  spherical,  19. 

Proper  motions,  240. 


Quadrant,  47. 
Quadrature,  136. 

Radiant  points,  231. 
Rate  of  clock,  23. 
Refraction,  50;  astronom 

general  laws,  52;  effecl 
Resisting  medium,  219. 
Reticule,  27. 
Retrograde,  motion,  174, 
Rings  of  Saturn,  196;  di 

ence  of,  198. 

Saros,  157. 

Satellites,  elements  of,  29' 

Saturn,  196;  rings  of,  196, 

Seasons,  95. 

Sextant,  39;  prismatic,  47 

Shadow,  of  earth,  148;  c 
152. 

Signs  of  zodiac,  87. 

Sirius,  light  of,  242;  motioi 

Solar  system,  1;  orbit  of, 
ments  of,  291. 

Solstices,  87. 

Spectra,  solar,   103;  stell 
nebular,  263. 

Spectroscope,  38. 

Sphere,    celestial,    1;   par 
right  and  oblique,  7. 

Spots,  solar,  100;  observa 
297. 

Star  signals,  82. 

Stars,  circumpolar,  5;  fix 
number  of,  241;  ma£ 
241;  constitution,  246;  ( 
246;  differential  obsei 
249;  real  dimensions,  2 
iable  and  temporary,  2,1 
ble  and  binary,  255; 
257;  examples  of  varial 
of  binary,  303. 

Stationary  points,  175,  18( 


INDEX 


309 


i  and  new,  113. 
tance  of,  90;  dimensions, 
tation,  100;  constitution, 
pparent  motion,  106;  first 
108;  second  mean,  110; 
and  density,  116;  size 
red  with  stars,  251;  mo- 
Q  space,  266;  elements, 
x>ts,  100. 

1  period,  of  moon,  138; 
lets,  180. 

method  of  finding  lati- 
T6. 

h,  used  in  determining 
ide,  81. 

ic  comets,  213. 
i  comets,  219. 
te,  47. 

K);  daily  inequality,  162; 
I  laws,  163;  influence  of 
>4;  spring  and  neap,  165; 
g  and  lagging,  165;  tidal 
165;  establishment,  166; 

lines,  167;  height,  167; 
lily,  169;  in  lakes,  169. 
lar  and  sidereal,  12,  106; 
1  and  right  ascension, 
•eenwich,  79;  local  times 
erent  meridians,  84;  as- 
lical  and  civil,  111. 


Torsion  balance,  65. 

Trade-winds,  70. 

Transit  instrument,  25. 

Transit,  11;  of  inferior  planets  298. 

Triangle,  astronomical,  15. 

Triangulation,  61. 

Twilight,  98. 

Umbra,    of   solar    spots,    101;   of 

eclipses,    148. 
Universal  instrument,  47. 
Uranus,  200;  satellites  of,  294. 
Ursa,  major,  243;  minor,  243. 

Vanishing  points  and  circles,  144. 

Venus,  relative  distance  from  the 
sun  arid  earth,  88;  transit  of, 
89,  182,  198;  description  of,  182. 

Vernier,  44. 

Vertical,  circles,  10;  prime,  10. 

Weight,  in  different  latitudes,  66; 

on  the  sun,  117. 
Year,  sidereal,   86;  tropical,   112, 

123;  anomalistic,  124. 

Zenith,  9;  geographical  and  geo- 
centric, 78. 

Zenith  telescope,  47;  use  of,  76. 
Zodiac,  signs  of,  87. 
Zodiacal  light,  104. 


IiMrff  / 


Wiley  Special  Subject  Catalogues 

For  convenience  a  list  of  the  Wiley  Special  Subject 
Catalogues,  envelope  size,  has  been  printed.  These 
are  arranged  in  groups  —  each  catalogue  having  a  key 
symbol.  (See  special  Subject  List  Below).  To 
obtain  any  of  these  catalogues,  send  a  postal  using 
the  key  symbols  of  the  Catalogues  desired. 


1 — Agriculture.     Animal  Husbandry.    Dairying.    Industrial 
Canning  and  Preserving. 

2— Architecture.       Building.      Concrete  and  Masonry. 

3 — Business  Administration  and  Management.     Law. 

Industrial  Processes:  Canning  and  Preserving;    Oil  and  Gas 
Production;  Paint;  Priming;  Sugar  Manufacture;  Textile. 

CHEMISTS 

4a  General;  Analytical,  Qualitative  and  Quantitative;  Inorganic; 

Organic. 

4b  Electro-  and  Physical;  Food  and  Water;  Industrial;  Medical 
and  Pharmaceutical;  Su  .ir. 
iiii!*n£ir  v>*r,/I  .1 

CIVIL  ENGINEERING 

5a  Unclassified  and  Structural  Engineering. 

5b  Materials  and  Mechanics  of  Construction,  including;  Cement 

and    Concrete;    Excavation    and    Earthwork;    Foundations; 

Masonry. 

5c   Railroads;  Surveying. 

3d  Dams;  Hydraulic  Engineering;  Pumping  and  Hydraulics;  Irri- 
gation Engineering;  River  and  Harbor  Engineering;  Water 
Supply. 


CIVIL  ENGINEERING—  Continued 

5e  Highways;  Municipal  Engineering;  Sanitary  Engineering; 
Water  Supply.  Forestry.  Horticulture,  Botany  and 
Landscape  Gardening. 


6 — Design.       Decoration.       Drawing:     General;     Descriptive 

Geometry;  Kinematics;  Mechanical. 

ELECTRICAL  ENGINEERING— PHYSICS 

7 — General  and  Unclassified;  Batteries;  Central  Station  Practice; 
Distribution  and  Transmission;  Dynamo-Electro  Machinery; 
Electro-Chemistry  and  Metallurgy;  Measuring  Instruments 
and  Miscellaneous  Apparatus. 


8 — Astronomy.      Meteorology.      Explosives.      Marine    and 
Naval  Engineering.     Military.     Miscellaneous  Books. 

MATHEMATICS 

iSQP'T'l    i 

9 — General;    Algebra;  Analytic  and  Plane   Geometry;   Calculus; 
Trigonometry;  Vector  Analysis. 

MECHANICAL  ENGINEERING 

lOa  General  and  Unclassified;  Foundry  Practice;  Shop  Practice. 
lOb  Gas  Power  and    Internal   Combustion  Engines;  Heating  and 

Ventilation;  Refrigeration. 
lOc  Machine  Design  and  Mechanism;  Power  Transmission;  Steam 

Power  and  Power  Plants;  Thermodynamics  and  Heat  Power. 
1 1 — Mechanics.  ______ 

12 — Medicine.  Pharmacy.  Medical  and  Pharmaceutical  Chem- 
istry. Sanitary  Science  and  Engineering.  Bacteriology  and 

Biology. 

MINING  ENGINEERING 

13 — General;  Assaying;  Excavation,  Earthwork,  Tunneling,  Etc.? 
Explosives;  Geology;  Metallurgy;  Mineralogy;  Prospecting* 
Ventilation. 

14 — Food  and  Water.  Sanitation.  Landscape  Gardening. 
Design  and  Decoration.  Housing,  House  Painting. 


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